Vectors Work
Short Description
Vectors +2...
Description
p × [( x − q ) × p ] + q × [( x − r ) × q ] + r × [( x − p ) × r ] = 0 p = q = r = k r × [ x − p ) × r ] = 0 p × [( x − q ) × p ] + q [( x − r ) × q ] a = b = c = 1 66 (c × a ).(b × a ) (c × b).( a × b) l= 2 ( a × b) 2 (b × a ) a × b ) + β (b × c ) + γ ( c × a ) = 0
2 c b a a × (b × 2c ) = b a , b , c β × γ ) = pα + qβ + rγ α γ 3
β = 2i − J + k α O a × b 2 a × b 2 ( BD × AC ).(OD × OC ) DB OA OB OA OA = a , OB = b , OC = 2a + 3b , OD = a − 2b p + q < 1 p and q
A(1, 0, 2)
B(–2, 1, 3)
TORS
D
C(2, 1, 1)
1 1 1 1 1 − 2 cos 1 AD = − i − J − k , AD = , , 3 3 3 3 1 + 2 cos θ 1 + 2 cos 2 1 + 2 cos r = a + λb 1 2 cosθ −1 − , , 1 + 2 cos θ 1 + 2 cos θ 1 + 2 cos θ a a.a b.a a1 b a.b b.b = b1 c a.c b.c c1
a2 b2 c2
a 3 i a1 b3 × j a 2 c3 k a3
b1 a b c − 11iˆ + 10 Jˆ + 2kˆ c c a cˆ = b2 = 0 a.a a.b a.c = 0 15 b3 b.a b.b b.c
iˆ + cos( β − γ ) ˆj + akˆ iˆ + ˆj + cos(γ − β )kˆ iˆ + cos( β − α ) ˆj + cos(γ − α )kˆ 11 r .(3iˆ − ˆj + k 2 1 1 R n t n t r R t r R n r R t r R R b : c O, b , c 1 − r 2i − J − k r + (t × n ) 2 2 r .(t × n ) a , b and c kˆ kˆ kˆ a × b ) 2 c = (c × b ).( a × b )a + (a × c ).( a × b )b b a a , b , c a × (b × c ) a = b + c
iˆ + 2 ˆj + kˆ r .a = 0 b r − c r a.b ≠ 0 a.a b .a c .a →
→
a x b)
a.b b .b c .b
a.c c × d ) + (b × c ).( a × d ) + (c × a ).(b × d ) = 0 b .c c .c
u ⊥ v (u × v ).w ≤ 1 / 2 v w + w × u w u & v a a OE .CD = 0 (u1 × u 2 ).w ≤ 1 / 2
w + w × u1 = u 2 ω u 2 u1 b a c c b a c c b a a × b + b × c = pa + qb + rc ˆ − 11i − 10 J + 2k c c= 15
w + w × u1 = u 2 ω u 2 u1 b a c c b a c c b a a × b + b × c = pa + qb + rc ˆ − 11i − 10 J + 2k c= 15 cc c '×a' a '×b' 1 a.b b'×c' b= c= r= 2 a + kb + a × b 2 [ a ' b' c ' ] [ a' b' c' ] k +a k [ a ' b' c ' ] b × ( a × b) , ( a × b) 2
( a × b) × a b' = , ( a × b) 2
a × b a × b c × a b × c AB PQ CD AB bˆ − (aˆ.b ) aˆ c' = (a × b) 2 [abc ] [abc] [abc ]
→→ 3 (aˆ × bˆ) aˆ b aˆ b & c (e .c )a = c c b a × (b × c ) + (a.b )b (a − b ).c r = b + tc a → b a b b+ 2 c − a c → →
→
a b = 4iˆ − ˆj + 3kˆ a = 3iˆ + 6 ˆj − 2kˆ
→ → → ˆ ˆ ˆ 1 1 1 + + = 1 a # b # c #1 i + j + ck 1− a 1− b 1− c
→ aiˆ + ˆj + kˆ, iˆ + bˆj + kˆ 5 iˆ + 2 ˆj + 2 kˆ 2iˆ + 2 ˆj + kˆ n→ + n→ + n→ + n→ = 0 → 4iˆ + 3 ˆj → c b b 1 2 3 4 3 3 3
( x − a) 2 ( y − a) 2 ( z − a) 2
( x − b) 2 ( y − b) 2 ( z − b) 2
→ → → → → → → → → → → → → → ( x − c) 2 → a, b, c 5 a + 3 b + 5 c 3 a + b + 2 c a − b − c a + 2 b + c ( y − c) 2 ( z − c) 2
BASIC CONCEPTS OF VECTORS: In mathematics, physics, and engineering, a Vector (sometimes called a geometric or spatial vector) is a geometric quantity having magnitude (or length) and direction expressed numerically as ordered list of coordinates [x, y, z]. A vector is an object that is an input to, or output from vector functions as per vector algebra. A Vector is a directed line segment, or arrow, connecting an initial point with a terminal point. Is a vector with initial point A and terminal point B. Technically, the [x, y, z] components of vector
are equal to the vector difference
minus .
DEFINITION: A scalar is a quantity, which has only magnitude but does not have a direction. For example time, mass, temperature, distance and specific gravity etc. are scalars. A Vector is a quantity which has magnitude, direction and follow the law of parallelogram (addition of two vectors). For example displacement, force, acceleration are vectors. a a (a) There are different ways of denoting a vector: or or a are a , b, c
different ways. We use for our convenience vectors, and a, b, c to denote their magnitude. (b) gives
A vector may be represented by a line segment OA and arrow direction of this vector. Length of the line segment gives the magnitude of the
vector. A
O
H e r e O is th e in itia l p o in t a n d A is th e te r m in a l p o in t o f O A
CLASSIFICATION OF VECTORS: (i)
etc. to denote
Equal vectors
Two vectors are said to be equal if and only if they have equal magnitudes and same direction.
(ii)
A
B
C
D
A B = C D A s w e ll a s d ire c tio n is s a m e
Zero Vector (null vector): A vector whose initial and terminal points are same, is called
Zero vector. For example
AA
.Such
vector has zero magnitude and no direction, and denoted by AB + BC + CA = AA
Or
AB + BC + CA = 0
B
Like and Unlike Vectors: Two vectors are said to be (a) Like, when they have same direction. (b) Unlike, when they are in opposite directions. •
and –
a
are two unlike vectors as their a
(iv)
a
a as aˆ = a
aˆ
. Therefore .
Parallel vectors: Two or more vectors are said to be parallel, if they have the same support or parallel support. Parallel vectors may have equal or unequal magnitudes and direction may be same or opposite. As shown in figure a
O C E
(vi)
a
directions are opposite, and 3 are like vectors. Unit Vector: A unit vector is a vector whose magnitude is unity. We
write, unit vector in the direction of (v)
.
C
A
(iii)
0
A b c
B D
OP
Position Vector: If P is any point in the space then the vector is called position vector of point P, where O is the origin of reference. Thus for any points A and B in the space,
AB = OB − OA
(vii) Coinitial vectors: Vectors with same initial point are called coinitial vectors. OA, OB, OC
As shown in figure
and
OD
are CoInitial vectors.
D
A d
C
a
O
c
B
b
.
ADDITION OF TWO VECTORS: Let
OA = a , AB = b
Here the
c
and
OB = c
.
is sum (or resultant) of vectors initial
a
to the terminal
point of
b
represents vector
and
b
. It is to be noticed that
b
point
coincides with the terminal point of point of
a
a+b
a
of
and the line joining the initial
in magnitude and direction.
PROPERTIES: (i)
(ii) (iii) vectors) (iv) vectors)
a +b = b+a
a + ( b + c ) = (a + b ) + c  a + b ≤ a  +  b 
 a + b  ≥  a  −  b 
(So, Vector addition is commutative) (So, Vector addition is associative) (equality holds when
(equality holds when
a
a
and
and
b
b
are like
are unlike
a +0 =a =0+a a + ( −a ) = 0 = ( −a ) + a
(v) (vi)
.
MULTIPLICATION OF VECTOR BY SCALARS : If
a
is a vector and m is a scalar, then m
a
is a vector parallel to
a
a
whose modulus is  m  times that of . This multiplication is called Scalar a b a Multiplication. If and are vectors and m, n are scalars, then: m( ) a a a a a = ( )m = m m(n) = n(m) = (mn) b a a a a b a (m + n) =m +n m( + ) = m + m . LINEAR COMBINATIONS: Given
a
finite
set
r = xa + yb + zc + ........
of
vectors
a , b , c ,......
then
is called a linear combination of
the
vector
a , b , c ,......
for
any x, y, z..... . We have the following results: (i)
(ii)
If
a,b
are
nonzero,
xa + yb = x' a + y' b ⇒ x = x' ; y = y'
(iv)
vectors
then
. a,b
Fundamental Theorem: Let be nonzero , non collinear vectors a,b r . Then any vector coplanar with can be expressed uniquely as a linear combination of
(iii)
noncollinear
a,b
xa + yb = r that a, b,c If are nonzero, noncoplanar vectors then: xa + yb + zc = x' a + y' b + z' c ⇒ x = x' , y = y' , z = z' .. x1 , x 2 ,...... x n
If
&
∈
i.e. There exist some unique x, y R such
are n non zero vectors, & k , k , .....k are n scalars 1 2 n if the linear combination
k1x1 + k 2 x 2 +........ k n x n = 0 ⇒ k1 = 0, k 2 = 0..... k n = 0
then
we
say
that
vectors are Linearly Independent Vectors . (v)
x1 , x 2 ,...... x n
If Linearly
are not Linearly Independent then they are said to be Dependent vectors. i.e. if.
k1x1 + k 2 x 2 +........ k n x n = 0 ⇒ k1 = 0, k 2 = 0..... k n = 0
at least one then
x1 , x 2 ,...... x n
Example: Show that the vectors coplanar (Where Solution: Let
a , b, c
& if there exists
Kr ≠ 0
are said to be Linearly Dependent.
5a + 6b + 7c, 7a − 8b + 9c
&
3a + 20b + 5c
are
are three noncoplanar vectors.).
A = 5a + 6b + 7 c B = 7a − 8b + 9c
,
and
C = 3a + 20b + 5c
.
A, B
x A + yB + zC = 0 C and are coplanar. This indicates that must have a real solution for x, y, z other than (0, 0, 0).
Now,
x (5a + 6b + 7 c) + y(7a − 8b + 9c) + z(3a + 20b + 5c) = 0 (5x + 7 y + 3z)a + (6x − 8y + 20z)b + (7 x − 9 y + 5z)c = 0
5x + 7y + 3z = 0 6x – 8y + 20 z = 0 7x + 9y + 5z = 0 (As 5
7
a , b, c
are noncoplanar vectors)
3
6 − 8 20 = 0 7 9 5
Now D = So the three linear simultaneous equation in x, y and z have a nontrivial solution. Hence the given vectors are coplanar vectors. COLLINEARITY AND COPLANARITY OF POINTS:
(a)
The necessary and sufficient condition for three points with position
vectors
a , b, c
to be col
not all zero such that Where (b)
is that there exist three scalars x, y, z, linear xa + yb + zc = 0
x+y+z=0
.
The necessary and sufficient condition for four points with position
vectors
a , b, c
and
d
to be coplanar is that there exist scalars x, y, z, u,
xa + yb + zc + ud = 0
x+ y +z+u = 0
not all zero, such that where . Example: Let 'O' be the point of intersection of diagonals of a parallelogram ABCD. The points M, N, K &P are the mid points of OA, MB, NC and KD respectively. Show that N, O and P are collinear. Solution: D
C P K O
M
A
N
a +b a a + 2b M≡ , N≡ 2 = 2 2 4 Now, a+b −a 2 b − 3a K≡ 4 = 2 8 2b − 3a −b+ − 6 b − 3a 8 P≡ = 2 16 uuu 3 OP = − (2b + a ) 16
B
uuu 1 1 uuu ON = (a + 2b ) = − OP 4 6
( )
. Hence the points N, O& P are collinear. SECTION FORMULA: Let A, B & C be three non collinear points in space having the a, b r position vectors and . ( x 1, y 1, z 1)
A
n C
a
m
r O
Let
(x , y, z )
B ( x 2, y 2, z 2) b
AC n = CB m
mAC=nCB
m AC = n CB
. (As vectors are in same direction) OA + AC = OC ⇒ AC = r − a Now, r + CB = b ⇒ CB = b − r ma + nb r= m+n
Using (i), we get
.
ORTHOGONAL SYSTEM OF UNIT VECTORS: Let OX, OY and OZ be three mutually perpendicular straight lines. Given any point P(x, y, z) in space, we can construct the rectangular parallelepiped of which OP is a diagonal and OA = x, OB = y, OC = z. Here A, B, C are (x, 0, 0), (0, y, 0) and (0, 0, z) respectively and L, M, N are (0, y, z), (x, 0, z) and (x, y, 0) respectively.
Y B
N P (x, y, z)
O
X
A
Let
ˆi , ˆj, kˆ
We have
M
C
Z
denote unit vectors along OX, OY and OZ respectively. r = OP = xˆi + yˆj + zkˆ
as
OA = xˆi , OB = yˆj
and
OC = zkˆ
.
ON = OA + AN uuu uuu uuu OP = ON + NP. So, uuu uuu uuu uuu uuu uuu uuu uuu OP = OA + OB + OC NP = OC , AN = OB
(
)
r xˆi + yˆj + zkˆ r= = = ˆi + mˆj + nkˆ 2 2 2 r x +y +z ⇒ r = rˆi + mrˆj + nrkˆ
.
MULTIPLICATION OF VECTORS: SCALAR PRODUCT OF TWO VECTORS (DOT PRODUCT): a
b
The scalar product, of two nonzero vectors and is defined as  a   b  cos θ θ .where is angle between the two vectors, when drawn with same initial point. Note that If at least one of PROPERTIES : (i) (ii)
0≤θ≤π
a
b
.
and is a zero vector, then
a.b = b.a (scalar product is commutative) 2 2 2 a = a.a =  a  = a
a.b
is defined as zero.
(iii) (iv)
(ma ).b = m(a.b) = a (mb) a.b −1 θ = cos  a . b  a.b = 0 ⇔ a
(v) Vectors nonzero vectors]. (vi) (vii) (viii) (ix)
(xi)
and
b
a b
are perpendicular to each other. [ ,
ˆi.ˆj = ˆj.kˆ = kˆ.ˆi = 0 a.( b + c) = a.b + a.c (a + b).(a − b) =  a 2 −  b 2 = a 2 − b 2
Let
a = a1ˆi + a 2 ˆj + a 3 kˆ, b = b1ˆi + b 2 ˆj + b 3 kˆ
Then (x)
(where m is a scalar)
,
a.b = (a 1ˆi + a 2 ˆj + a 3 kˆ ).(b1ˆi + b 2 ˆj + b 3 kˆ )
.
Maximum value of
b a b a . =   
Minimum value of
b a b a . =–   
a
(xii) Any vector can be written as,
a
=
ej ej d i
a. i i + a. j j + a. k k
.
Algebraic projection of a vector along some other vector:
are
a.b ON = OB cos θ =  b  = aˆ.b  a  b  B b
O
θ
a
A
Example: Prove that the angle in a semicircle is a right angle. Solution: Let O be the centre and AB the bounding diameter of the semicircle. Let P be any point on the circumference. With O as origin. Let
OA = a , OB = −a
and
OP = r
.
P
B
O
A
As OA = OB = OP, each being equal to radius of the semicircle. AP = r − a
and
BP = r − (−a ) = r + a
AP.BP = (r − a ).(r + a ) = r 2 − a 2
2 2 = OP – OA = 0
AP and BP are perpendicular to each other, i.e. VECTOR (CROSS) PRODUCT: The vector product of two nonzero vectors
a
and
b
∠APB = 900
.
, whose modules
are a and b respectively, is also a vector whose modulus is
ab sin θ
,
where is
θ(0 ≤ θ ≤ π)
n
b
a
the angle between vectors and . The direction is
that of a vector perpendicular to both righthanded orientation.
a
b
& , such that
a , b, n
are in
b θ
O
a
a × b =  a   b  sin θ nˆ
.
PROPERTIES: (i) (ii) (iii)
a × b = −( b × a ) ( ma ) × b = m(a × b) = a × (mb) (Where m is a scalar) a×b = 0 ⇔ b a
vectors
and
are parallel.
(provided
a
and
b
are nonzero
vectors). (iv) (v) (Vi)
ˆi × ˆj = ˆj × ˆj = kˆ × kˆ = 0 ˆi × ˆj = kˆ = −(ˆj × ˆi ), ˆj × kˆ = ˆi = −(kˆ × ˆj), kˆ × ˆi = ˆj = −(ˆi × kˆ ) a × (b + c) = a × b + a × c a = a 1ˆi + a 2 ˆj + a 3 kˆ b = b1ˆi + b 2 ˆj + b 3 kˆ
(vii) Let
and
ˆi ˆj a × b = a1 a 2 b1 b 2
(Viii) . (ix)
a×b sin θ =  a  b 
Area of triangle =
, then
kˆ a3 b3
=
ˆi(a b − a b ) + ˆj(a b − a b ) + k(a ˆ b −a b ) 2 3 3 2 3 1 1 3 1 2 2 1
1 1 1 ap = ab sin θ =  a × b  2 2 2
.
B b O
(x)
P
θ
A
a
Area of parallelogram =
C
B
b
axb≠bxa
p
θ
O
(xi)
ap = ab sin θ = a × b 
A
a
.
(not commutative)
axb a & b is ± axb
(xii) Unit vector perpendicular to the plane of . (xiii) A vector of magnitude ‘r’ & perpendicular to the plane of r axb a & b is ± axb
(
)
. (xiv) Area of any quadrilateral whose diagonal vectors are
d1 & d 2
1 d1 x d 2 2
by . (xv) LaGrange’s
Identity:
2 2 a .a a & b ;(a x b) 2 = a b − (a . b) 2 = a .b
For
a .b b.b
any
two
is given
vectors
.
Example: If a, b, c be three vectors such that a + b + c = 0, prove that a × b = b × c = c × a and Solution: Let
BC, CA , AB
deduce
sin A sin B sin C = = a b c
(Sine rule).
represent the vectors a, b, c respectively.
Then, we have A
π A
b
c
π C
B C
a
π B
a + b + c = 0, ==> c =  (a + b) ==> b × c = b × ( a  b) =b×a=a×b Similarly, c×a=a×b Hence,
bc sin(π − A ) = ca sin( π − B) = ab sin(π − C)
==> b × c = c × a = a × b
==> bc sin A = ca sin B = ab sin C. Hence,
bc sin(π − A ) = ca sin( π − B) = ab sin(π − C)
.
SCALAR TRIPLE PRODUCT (BOX PRODUCT): The scalar triple product of three vectors
(a × b).c
the between Let
=
 a   b   c  sin θ cos φ
axb & c
a, b
θ
c
& is defined as
where is the angle between angle
. It is also denoted as
[a b c]
.
a = a1iˆ + a2 ˆj + a3 kˆ, b = b1iˆ + b2 ˆj + b3kˆ, c = c1iˆ + c2 ˆj + c3kˆ
Then
ˆi ˆj a × b = a1 a 2 b1 b 2
kˆ a a 3 = ˆi 2 b2 b3
a 3 ˆ a1 a 3 ˆ a1 a 2 −j +k b3 b1 b 3 b1 b 2
.
a&b
φ
& is
a (a × b).c = c1 2 b2
a1 a 2 a3 a1 a 3 a1 a 2 − c2 + c3 = b1 b 2 b3 b1 b 3 b1 b 2 c1 c 2
a3 b3 c3
.
(a × b).c = (b × c).a = (c × a ).b = −(b × a ).c = −(c × b).a = −(a × c).b Therefore . (a × b).c = (b × c).a = a.( b × c)
Note that
, hence in scalar triple product dot and cross are (a × b).c [a b c ] interchangeable. Therefore we indicate by . PROPERTIES: (i)
 (a × b).c 
represents the volume of the parallelepiped, whose adjacent
sides are represented by vectors Therefore three vectors a1 a 2 b1 b 2 c1 c 2
a3 b3 = 0 c3
a , b, c
,
(ii)
Volume of the tetrahedron =
(iii)
[a + b c d ] = [a c d ] + [ b c d ] [a a b ] = 0
a , b, c
in magnitude and direction. [a b c ] are coplanar if = 0. i.e.
1  [( a b c]  6
.
(IV) . (v) In a scalar triple product the position of dot & cross can be interchanged i.e. a . ( b x c) = (a x b). c OR [ a b c ] = [ b c a ] = [ c a b ]
(vi)
a . (b x c) = − a .( cx b) i. e. [ a b c ] = − [ a c b ] b a c
(vii) If = a i+a j+a k; 1 2 3
= b i+b j+b k & = c i+c j+c k then 1 2 3 1 2 3
a1 a 2 a 3 [a b c] = b1 b 2 b 3 c1 c2 c3
In general, i
a = a 1 l + a 2 m + a 3 n b = b1 l + b 2 m + b 3 n
;
a1 a b c = b1
a2
a3
b2
b3
c1
c2
c3
[ ]
.
[ l m n]
&
c = c1 l + c2 m + c3 n
then
; where
a,b,c
, m & n
⇔ [a b c] = 0
are noncoplanar vectors .
(viii) If are coplanar . (ix) Scalar product of three vectors, two of which are equal or parallel is 0 [a b c] = 0 i.e. . a,b,c Note : If are noncoplanar then [a b c] > 0 & [a b c] < 0
for left handed system
(x) (xi) (xii)
for
handed
system
[ a + b
]
[ a b c]
.
[i j k] = 1. [ K a b c ] = K[ a b c ]
.
[(a + b) c d ] = [ a c d ] + [ b c d ]
(xiv) Remember that: .
[ a − b
.
]
b− c c−a
=0
b+ c c+a
&
VECTOR TRIPLE PRODUCT: If
right
a,b,c
The vector triple product of three vectors at least a,b,c
is
a
zero
vector
b
c
and are
=2
is defined as
collinear
a × ( b × c)
vectors
. one a
or is
perpendicular b
a × (b × c) = 0
c
and , only then nonzero vector
to
a × ( b × c)
. In all other cases in the
both will be a a
plane of noncollinear vectors and perpendicular to the vector . a × (b × c) = λb + µc λ µ Thus we can take , for some scalars & . Since a ⊥ a × ( b × c) , a.(a × (b × c)) = 0 ⇒ λ(a.b) + µ(a.c) = 0 ⇒ λ(a.c)α, µ = −(a.b)α
α
.
a × (b × c) = (a.c)b − (a.b)c
a
Example: For any vector , prove that Solution:
for same scalar
ˆi ×(a × ˆi) + ˆj×(a × ˆj) + kˆ × (a×k) ˆ = 2a
.
[ˆi × (a × ˆi )] + [ˆj × (a × ˆj)] + [kˆ × (a × kˆ )]
= =
[( ˆi.ˆi )a − (ˆi.a )ˆi ][( ˆi.ˆj)a − (ˆj.a )ˆj] + [( kˆ.kˆ )a − ( kˆ.aˆ )kˆ ] a − (ˆi.a )ˆi + a − (ˆj.a )ˆj + a − (kˆ.aˆ )kˆ [
=
ˆi.ˆi = ˆjˆj = kˆkˆ = 1]
3a − [( ˆi.a )ˆi + (ˆj.a )ˆj + (kˆ.aˆ ) kˆ ]
Let
a = a 1ˆi + a 2 ˆj + a 3 kˆ
. Then
ˆi a = ˆi (a ˆi + a ˆj + a kˆ ) = a ˆi 2 + a (ˆi.ˆj) + a (ˆi.kˆ ) = a (1) + a (0) = a 1 2 3 2 2 3 1 2 1
Similarly,
ˆj.aˆ = a , kˆ.a = a 2 3
3a − (a 1ˆi + a 2 ˆj + a 3 kˆ ) = 3a − a = 2a
.
L.H.S. = R.H.S. RECIPROCAL SYSTEM OF VECTORS: a ′, b′
Let and
a, b
c
& be three noncoplanar vectors. Then the system of vectors c′
which satisfies
a.a ′ = b.b′ = c.c′ = 1
called
a.b′ = a.c′ = b.a ′ = b.c′ = c.a ′ = c.b′ = 0
and the
, is
a , b, c reciprocal system to the vectors . b× c c×a a×b a ′ = b′ = c ′ = [a b c ] [a b c ] [a b c ]
,
,
.
PROPERTIES: (I) (II) is
a.b′ = a.c′ = b.a ′ = b.c′ = c.a ′ = c.b′ = 0
The scalar triple product [a b c] of three noncoplanar vectors a, b, c the reciprocal of the scalar triple product formed from reciprocal system.
Example: Solve the vector equation:
r × b = a × b , r .c
b
is not perpendicular Solution: We are given;
to .
r ×b = a×b (r − a) × b = 0 ( r − a) b Hence and are parallel r −a = t b r .c
And we know
Or
. . . (i)
= 0,
Taking dot product of (i) by r .c − a.c
=
c
t (b.c)
a.c t (b.c ) – = a.c b.c
t=–
r
from (i) and (ii) solution of is r
=
a
–
a.c b.c b
= 0 provided that
.
we get
. . . (ii)
c
KEY CONCEPTS · If
are any `n' vectors and
are scalars, then
known as linear combination (LC) of the vectors · If
are any `n' vectors and
is
.
are scalars. IF
and
atleast one of is not equal to zero, then the vectors be linearly dependent(LD)
are said to
· If
are any `n' vectors and
are scalars. If
and
if all are zero then the vectors are said to be linearly independent (L I). · Two collinear vectors are always linearly dependent. · Three coplanar vectors are always linearly dependent. · If
are any two vectors then
· If
are any two vectors then
· If
are any two vectors then
is a vector perpendicular to both
and · If
are any three vectors then scalar tripple product between the
vectors is defined as
and denoted by
· · · In general · · If
=0 then the vectors
· If the vectors
are said to be coplanar.
are coplanar then
are also coplanar.
STATEMENTS AND REASONING EXAMPLES: 1.
STATEMENT 1: If
x&v
are unit vectors inclined at an angle and
is a unit vector bisecting the angle between them, then
x
x+v x= α 2 cos 2
Because STATEMENT2: If ABC is an isosceles triangle with AB = AC = 1, then vector representing bisector of angle A is given by
uuu AD
=
AB + AC 2
(A) Statement1 is True, Statement2 is True; Statement2 is a correct explanation for Statement1 (B) Statement1 is True, Statement2 is True; Statement2 NOT a correct explanation for Statement1. (C) Statement1 is True, Statement2 is False (D) Statement 1 is False, statement2 is True. 1.
Option (a) is correct In an isosceles triangle ABC is which AB = AC, the median and bisector from A must be same line statement 2 is true. Now &
uuu x + v AD = 2
uuu 1 α  AD  = 2 cos 2 2 2
, So
uuu  AD 
= cos
α 2
unit vector along AD i.e. x is given by 2.
uuu AD x = uuu  AD 
=
STATEMENT 1: The points with position vectors 4a − 7b + 7c
a − 2b + 3c, − 2a + 3b − c
,
are collinear.
because STATEMENT2: The position vectors are linearly dependent vectors.
a − 2b + 3c,
2
a + 3b − c, 4a − 7b + 7c
(A) Statement1 is True, Statement2 is True; Statement2 is a correct explanation for Statement1 (B) Statement1 is True, Statement2 is True; Statement2 NOT a correct explanation for Statement1. (C) Statement1 is True, Statement2 is False (D) Statement 1 is False, statement2 is True. 1 (
a − 2b + 3c) + λ 2 (−2a + 3b − c) + λ 3 (4a − 7b + 7c) = 0
a, b & c
equating coefficients of
both sides we will get values of 1, 2 & 3 such that 1 + 2 + 3 = 0. Which is the condition for linearly dependent vectors & all for collinearity of the points. ‘a’ is correct.
3.
STATEMENT 1: If
a, b, c
then the angle between
are three unit vectors such that
a &b
1 a × (b × c) = b 2
is /2
because STATEMENT2: If
1 a × (b × c) = b, 2
then
a .b
= 0.
(A) Statement1 is True, Statement2 is True; Statement2 is a correct explanation for Statement1 (B) Statement1 is True, Statement2 is True; Statement2 NOT a correct explanation for Statement1. (C) Statement1 is True, Statement2 is False (D) Statement 1 is False, statement2 is True. a × (b × c) b − (a.b)c
whereas
a × (b × c)
comparing
a.b
=
=0
1 b 2
So
a
is perpendicular to
b
‘a’ is correct. 4.
STATEMENT 1: In ABC, because STATEMENT2: If
uuu uuu uuu AB + BC + CA
uuu uuu OA = a, OB = b
=0
uuu AB = a + b
then
(A) Statement1 is True, Statement2 is True; Statement2 is a correct explanation for Statement1 (B) Statement1 is True, Statement2 is True; Statement2 NOT a correct explanation for Statement1. (C) Statement1 is True, Statement2 is False (D) Statement 1 is False, statement2 is True. (C) In ABC
uuu uuu uuu uuu AB + BC = AC = − CA
uuu uuu uuu OA + AB = OB
uuu AB
+
uuu BC

uuu CA
=
u O
is triangle law of addition
Hence statement1 is true statement2 is false.
5.
STATEMENT 1: p = 9/2 and q = 2.
a = 3i + pj + 3k
because STATEMENT2:
If
and
a = a1 i + a 2 j + a 3 k
b = 2i + 3 j + qk
and
are parallel vectors it
b = b1 i + b 2 j + b3k
are parallel
a1 a 2 a 3 = = b1 b2 b3
(A) Statement1 is True, Statement2 is True; Statement2 is a correct explanation for Statement1 (B) Statement1 is True, Statement2 is True; Statement2 NOT a correct explanation for Statement1. (C) Statement1 is True, Statement2 is False
(D) Statement 1 is False, statement2 is True. (A) 3 p 3 = = 2 3 q
from
a = a1 i + a 2 j + a 3k, b + b1 i + b 2 j + b 3k
a = 3i + pj + 3k
6.
and
b
=
2i + 3 j + qk
are parallel
a1 a 2 a 3 = = b1 b 2 b 3
2
3 p 3 = = 2 3 q
STATEMENT 1: The direction ratios of line joining origin and point (x, y, z) must be x, y, z because STATEMENT2: If P is a point (x,y, z) in space and OP = r then directions cosines of OP are
x y z , , r r r
(A) Statement1 is True, Statement2 is True; Statement2 is a correct explanation for Statement1 (B) Statement1 is True, Statement2 is True; Statement2 NOT a correct explanation for Statement1. (C) Statement1 is True, Statement2 is False (D) Statement 1 is False, statement2 is True. (A) 7.
STATEMENT 1: r = a + αb
and
because
The shortest distance between the skew lines
r = c + βd
is
 [a − c bd]   b×d 
STATEMENT2: Two lines are skew lines if three axist no plane passing through than (A) Statement1 is True, Statement2 is True; Statement2 is a correct explanation for Statement1 (B) Statement1 is True, Statement2 is True; Statement2 NOT a correct explanation for Statement1. (C) Statement1 is True, Statement2 is False (D) Statement 1 is False, statement2 is True. (B) A – Defn B – Defn 8.
STATEMENT 1: IF because STATEMENT2:
a.b
a.b = 0
a
= 0 either
b
a =0
or
b
a
= 0 or
b
=0
(A) Statement1 is True, Statement2 is True; Statement2 is a correct explanation for Statement1 (B) Statement1 is True, Statement2 is True; Statement2 NOT a correct explanation for Statement1. (C) Statement1 is True, Statement2 is False (D) Statement 1 is False, statement2 is True. (D)
9.
STATEMENT 1:
A× B = B× A
because A × B = A  B 
STATEMENT2: fingers curls from A to B
sin
n
, when is angle, when your
(A) Statement1 is True, Statement2 is True; Statement2 is a correct explanation for Statement1 (B) Statement1 is True, Statement2 is True; Statement2 NOT a correct explanation for Statement1.
(C) Statement1 is True, Statement2 is False (D) Statement 1 is False, statement2 is True. (D) 10.
STATEMENT 1: If the vectors coplanar, then 2 is equal to 16. because STATEMENT2: The vectors
a,b
2iˆ − ˆj + kˆ
and
c
,
ˆi + 2ˆj − 3kˆ
and
3iˆ − λˆj + 5kˆ
a, (b × c
are coplanar iff
are
)=0
(A) Statement1 is True, Statement2 is True; Statement2 is a correct explanation for Statement1 (B) Statement1 is True, Statement2 is True; Statement2 NOT a correct explanation for Statement1. (C) Statement1 is True, Statement2 is False (D) Statement 1 is False, statement2 is True. 2 −1 1 1 2 −3 3 λ 5
= 0 = 4
2 = 16 Ans. (A) 11.
STATEMENT 1: A line L is perpendicular to the plane 3x – 4y + 5z = 10 because 3
STATEMENT2: Direct on cosines of L be <
5 2
,−
4 5 2
,
1 2
.
(A) Statement1 is True, Statement2 is True; Statement2 is a correct explanation for Statement1 (B) Statement1 is True, Statement2 is True; Statement2 NOT a correct explanation for Statement1. (C) Statement1 is True, Statement2 is False (D) Statement 1 is False, statement2 is True. 1.lx + my + nz = P be the equation of a plane in the normal form.
D.N. of the plane 3x – 4y + 5z = 10 3
be < 3, 4, 5 > D.C <
5 2
,
−4 1 , 5 2 2
>
Ans. (A). 12.
STATEMENT 1: The value of expression
ˆi(ˆj × k) ˆ + ˆj(kˆ × ˆi) + k(i ˆ ˆ × ˆj) = 3
because STATEMENT2:
ˆi(ˆj × k) ˆ = [i.j.k] ˆ ˆ ˆ =1
(A) Statement1 is True, Statement2 is True; Statement2 is a correct explanation for Statement1 (B) Statement1 is True, Statement2 is True; Statement2 NOT a correct explanation for Statement1. (C) Statement1 is True, Statement2 is False (D) Statement 1 is False, statement2 is True. ˆi(ˆj × k) ˆ + ˆj(kˆ × ˆi) + k(i ˆ ˆ × ˆj)
=
13.
ˆi.iˆ + ˆj.jˆ + k.k ˆˆ
= 1 + 1 + 1 = 3.
Ans. (A)
STATEMENT 1: A relation between the vectors
a×b r = a.a
because STATEMENT2:
r, a and b
is
r ×a = b
r.a = 0
(A) Statement1 is True, Statement2 is True; Statement2 is a correct explanation for Statement1 (B) Statement1 is True, Statement2 is True; Statement2 NOT a correct explanation for Statement1. (C) Statement1 is True, Statement2 is False (D) Statement 1 is False, statement2 is True.
Since Q
We have
=(
b= r×a a × b = a × (r × a)
a.a) r − (a.r)a a×b r = a.a
=
(a .a) r Q a. r
=0
Ans. (A) EXCERSISE 1.
Show that the vectors I – j – 2k, 2i + 3J + k and 7i + 3J – 4k are coplanar.
2.
Show that the points P(), Q (), R (), S () are coplanar given that are non – coplanar . The vertices of triangle are A (2,3,0) B (3,2,1), C (4,1,0). Find the area of the triangle ABC and unit vector normal to the plane of this triangle. Let OACB be a //gm with O at the origin and OC a diagonal. Let D be the mid point of OA using vector methods, Prove that BD and CO intercept in the same ratio. Determine this ratio. D, E divide side BC and CA of a triangle ABC in the ratio 2 : 3 respectively. Find the position vector of the point of interception of AD and BE and the ratio in which this point divides AD and BE. 10/9, 15/4. If a, b and c be three nonzero vectors, number of two of which are collinear. If the vector a + 2b is collinear with c, and b + 3c is collinear with a then find the value of a + 2b + 6c = 0 If = 0 And the vectors X = (x2, x,1), Y = (y2,y,1), Z = (z2,z,1) are non coplanar, then the vectors (a2,a,1), (b2,b,1), (c2,c,1) are coplanar.
3.
4.
5.
6.
7.
8.
9.
Let = and i be two vectors to each other in the xy plane. Find all vectors in the same plane having projections 1 and 2 along and respectively. A line makes angle , , and with the diagonals of a cube, prove that cos2 + cos2 + cos2 + cos2 = 4/3.
Let Ar, (r = 1,2,3,4) be the areas of the faces of a tetrahedron. Let nr, be the out ward drawn normals to the respective faces with magnitudes equal to corresponding areas. Prove that 11. Find a vector of magnitude 5 units, coplanar with vectors 3i – j – k and I + j – 2k and to the vector . 12. If A (1,1,1) and C (0,1,1) are given vectors, then find a vector B satisfying A x B C and A.B = 3. () 13. If the vectors, and are coplanar and (), then prove that . 14. Two sides of triangle are formed by the vectors , . Find the angles. 15. Let a, b, c be three vectors such that a + b + c = 0, a = 3, b = 5, c = 7. Find the angle between and . 16. a = 3, and b = 4. Find the value of for which the vectors a + b and a  b are to each other. 17. The area of a //gm whose diagonals are given by a = 3i + j – 2k and b = i – 3j + 4k. 18. If the unit vectors and are inclined at an angle 2 and a – b is less than 1. then if 0 , find the range of . 19. The volume of the tetrahedron whose vertices are the points with position vectors I 6j + 10k, I – 3j + 7k, 5i – j + k and 7i – 4j + 7k is 11 cubic then find . 20. Determine the value of c, so that for all real x the vector cxi – 6j + 3k and xi + 2j + 2cxk make an acute angle with each other. 21. 1. Show that the distance of a point A() to the line is 10.
2. Find the scalars , iff = (4  2  sin)+ (2 – 1) and (where are non collinear. 22.
3. Let be a unit vector and be a non zero vector not parallel to . Find the angles of the triangle, two sides of which are represented by the vectors and . 23.
4. The p. v. of two pts A and C are 9i – j + 7k and 7i – 2j + 7k respectively. The pt. of intersection of vectors = 4i – j  3k and = 2i – j + 2k is P. If vector is to and CD and PQ = 15 units. Find P. v. of Q. 24.
1. Show that the segments joining vertices to the centriod of opposite faces of a tetrahedrown are concurrent. Hence find the position vector of the point of concurrence. 25.
Reciprocal System
Let a, b, c are set of non – coplanar vectors, the set of vectors a, b, c reciprocal to it is given by 26.
a = , b = , c = 27.
Find the system reciprocal to a, b, a b. a =
Find the set of vectors reciprocal to – i + J + k, i – J + k and i + J – k. 28.
If a, b, c are noncoplanar vectors, and a, b and c is the reciprocal system show that 29.
a=. Show that if k is a nonzero scalar and a and b are two vectors the soln. of the equation kr + r a = b is 30.
The vector – i + J + k bisects the angle between the vectors and 3i + 4j. Determine the unit vector 31.
along . Ans: Let a, b, c are unit vectors equally inclined at an angle to each other and for p, q, r are some scalars s.t . 32.
______ (1) Find p, q, r interms of . Let and be given non – zero and non collinear vectors and be a vector such that =  . Express in terms of &. 2. If and be two given non – collinear unit vectors and be a vector such that . Prove that . 3. In the triangle ABC, a point P is taken on the side AB such that AP: BP = 1:2 and a point Q is taken on the BC such that CQ: BQ = 2:1. If ‘R’ be the point of inter section of lines AQ and CP, using vector method find the area of ABC, if it is known that area of the ABC is one unit. 49/28 sq. unit. 1.
In ABC, D is the mid point of side AB and E is the centriod of CDA. If where 0 is the circumcentre of ABC, using vectors prove that AB = AC. 5. A non zero vector is to the line of inter section of the plane determined by the vectors i, i + J and the plane determined by the vectors i, – J + k. Find the angle between and i2J+ 2k. Q = cos1 1/3. 4.
P, Q are the mid points of the non – parallel sides BC and AD of a trapezium ABCD. Show that APD = CQP. 7. ABC and A1B1C1 are two coplanar triangles such that the perpendiculars from A, B, C to the sides B1C1, C1A1, A1B1 of the triangle A1B1C1 to the sides BC, CA, AB of the triangle ABC are also concurrent. 8. If two pairs of opposite edges of a tetrahedron are at right angles, then show that the third pair is also at right angle. Further show that for such a tetrahedron, the sum of the squares of each pair of opposite edges is the same. 9. If the from two vertices B & C to the opposite faces of a tetrahedron ABCD intersect, then BC in to AD. 10. Let be unit vectors. If is a vector such that = . Then prove that and that the equality holds iff . 6.
Q. 1. If the four points a, b, c, d are coplanar then show that [a b c] = [b c d] + [c a d] + [a b d]. Q.2. If a, b, c, d are any four vectors, then prove that (. (. Q.3. Show that, [a b b c c a] = [a b c]2 = Q. 4. If , find the vector which satisfies the eqns
() = 0 and .
Q. 5. Find a vector of magnitude 5 units, coplanar with vectors 3i – j – k and i + J –2 k and to the vector 2. Q.6. The value of [ab bc c – a] = 0. Q. 7. If d = (a b) + (b c) + v(c a) and [a b c] = 1/8. Then find the + + v in terms of a, b, c ,d.
Q. If , then show that = abc sin ( +). ( is angle between a and c, is angle between a + b.) Q. If are co – planar vectors and is not parallel to . Then prove that
(.
Q. Using vector method, find the ratio in which the bisector of an angle in a triangle divides the opposite side. Q. If the vectors i + cos(  ) j + cos(  d) where , , are different angles. If these vectors are coplanar. P. T. a is ind of , , . Cos(  )i + J + cos(  )
cos(  )i + cos(  )j + a
Q. If are unit vectors satisfying a + b + c = 0, then find a.b + b.c + c.a (a + b + c)2 = 0. Ans: –3/2 Q. Three non zero vectors p, q, r are pair wise non collinear. Further the vector p + q is collinear with r and q + r is collinear with p, find the sum of the vectors p, q and r. Q. In a ABC, are the position vectors of A, B, C. Prove that the internal bisector of angle A bisects the side BC internally in the ratio . (ii) Find the p. v. of the point in which a from B meets the internal bisector of angle A. (i)
Q. Solve the simultaneous vector eqns for . (  ). = 0, (). = 0, . = 1 &.n = 0 (All are non zero) consider  = + + () Ans: = Q. The vector i + 2J + 2k turns through a right angle and passing through the +ve x – axis on the way. Find the vector in its new position. Ans: 2 . Q. From the A(1, 2, 0), is drawn to the plane ) = 2, meeting it at the point P. Find the co – ordinates of point P and the distance AP. P (14/11, 21/11, 1/11), AP = 1/.
Q. cos(  ) cos(  ), where , and are different angles. If these vectors are coplanar show that is independent of , and . Q. The vector – i + j + k bisects the angle between the vectors and 3i + 4j. Determine the unit vector along . Q. If the vectors a, b, c are coplanar, show that Soln: Q. Show that the segments joining vertices to the centriod of opposite faces of a tetrahedron are concurrent. Hence find the position vector of the point of concumence. Eqn. of a line: Ans: and 1. The vertices of a ABC are A(1, 0, 2), B(–2, 1, 3) and C(2, 1, 1). Find the eqn. of the line BC, the foot of the from A to BC and the length of the .
2. If are two unit vectors inclined at an angle to each other. , if lies. (0, 2). Ans: (2/3, 4/3). 3. The vectors (al + al)i + (am + am)J + (an+ an)k (bl + bl)i + (bm + bm)J + (cn + cn)k.
form an equilateral triangle. Product of two determinants are coplanar are collinear are mutually . 4. If the length of is three times of the length of then is A. 7
B. 42
C.
is to
and
D. None
PRACTICE QUESTIONS Angle between Vectors 11. If
, then
(A) (B) (C) Projection along a Vector 12. Let of
and
and
whose projection on
(A) (B) Dot Product: 13. Let then
(D)
and
(C)
be three vectors, A vector in the plane is of magnitude
is
(D)
be two noncollinear unit vectors. If
and
is
(A) (B) (C) Modulus of Vectors:
(D)
14. If and c are unit vectors, then (A) 4 (B) 9 (C) 8 (D) 6 Parallel Vectors:
does not exceed
,
15. If
and
(A) (B) Parallel Vectors:
. If
(C)
, then d will be
(D)
16. The point of intersection of is (A) (B) (C) Section Formula:
and
and
, where
and
(D) None of these
17. Statement 1 : If I is the incentre of
then
Statement 2 : The position vector of centroid of is (A) STATEMENT1 is True, STATEMENT2 is True; STATEMENT2 is a correct explanation for STATEMENT1 (B)STATEMENT1 is True, STATEMENT2 is True; STATEMENT2 is NOT a correct explanation for STATEMENT1 (C)STATEMENT1 is True, STATEMENT2 is False (D)STATEMENT1 is False, STATEMENT2 is True Angle between Vectors: 18. Unit vectors and then may belong to
are inclined at an angle
(A) (B) (C) Modulus of Vector: 19. Let value
and
. If
,
(D) None of these
be the vectors such that
, if
and
then the
of is [IIT  95] (A) 47 (B) (C) 0 (D) 25 Vector Triple product: 20. Let
and
. If
is a vector such that
angle between and is , then (A) 2/3 (B) 3/2 (C) 2 (D) 3 Vector Product of four vectors:
[IIT  99]
and the
21. Let the vectors
and
be such that
planes determined by the pairs of vectors
. Let and
and
be
respectively, then the
angle between and is [IIT  99] (A) 0 (B) (C) (D) Basic concepts: 22. Matrix match type Column  I Column  II (A) Let
and then
(B) If
,
(p) 1
is
are three vectors of equal magnitude
(q)
and the angle between each pair of vectors in such that
then
is
(C) If
,
and
then
is
(D) If
are three non zero vectors such that
(r)
then the value of
(s) 0
may be`
(t) 8
Vector equation: PASSAGE: Consider a line and a plane 23. The angle between the line and the plane is (A) (B) (C) (D) 24. The position vector of the point of intersection of the line and the plane is (A) (B) (C) (D) 25. The position vector of the point on the given line whose distance from the plane is (A)
units is (B) (C)
(D) LEVELII MODEL QUESTIONS
1. Let to , then (A)
and a unit vector
(B)
(C)
2. If the vectors triangle ABC, then (A)
(D)
form the sides BC, CA and AB respectively of a
(B)
(C) 3. If
(D) and
are two unit vectors such that
to each other then the angle between (A)
be coplanar. If perpendicular
(B)
(C)
and
and is
(D)
4. Let there be two points A and B on the curve satisfying (A)
and
(B)
are perpendicular
in the plane OXY
then the length of the vector
(C)
is
(D)
5. If and D are four points in space satisfying then the value of k is (A) 2 (B) 1/3 (C) 1/2 (D) 1 6. If (A)
and (B)
(C)
and (D)
7. Given that (A)
and
(B)
8. The vector to (A)
(C)
(A)
and
and
is perpendicular
is
(D) None of these is twice its Ycomponent. If the magnitude of the vector
and it makes an angle of (B)
is :
(D) None of these
is perpendicular to
9. Xcomponent of is
then the angle between
(C)
. the angle between (B)
, then the angle between a and b is
(C)
with zaxis then the vector is : (D) None of these
10. If (A)
then the angle between (B)
(C)
and
is :
(D)
11. If the non zero vectors
are perpendicular to each other then the
solution of the equation,
is
(A)
(B)
(C) 12.
(D) None of these and
are mutually perpendicular unit vectors .
satisfying (A)
and (B)
13. If (A)
then
(C)
is equal to
(D)
then the value of (B)
(C)
is a vector
is
(D)
14*. If in a right angle triangle ABC, the hypotenuse then (A)
is equal to (B)
(C)
15. If
(D) and
are linearly dependent vectors and
then (A)
(B)
(C) (D) PRACTICE QUESTIONS
16. For three vectors which of the following expressions is not equal to any of the remaining three (A) (B) (C) (D) 17*. Which of the following expressions are meaningful ? (A) (B) (C) (D) 18. Let the unit vector a and b be perpendicular and the unit vector c be inclined at an angle (A) (C)
to both a and b. If (B) (D) None of these
then
19. The position vectors of the vertices of a quadrilateral ABCD are a,b,c and d respectively. Area of the quadrilateral formed by joining the middle points of its sides is (A)
(B)
(C) (D) 20. The volume of the tetrahedron, whose vertices are given by the vectors and with reference to the fourth vertex as origin, is (A) cubic unit (B) cubic unit (C) cubic unit (D) None of these 21. The coordinates of the foot of perpendicular drawn from point P (1 , 0 , 3) to the join of points
and
is
(A) (B) (C) (D) 22. Column I Column  II (A) The value of a for which the vectors
(p) 4
and are coplanar is (B) The area of a parallelogram having sides and (C)
(q)
is
and
for some non
(r)
zero vector , then the value of is (D) The volume of parallelopiped whose sides are given , and 23. Column  I Column II
is
(t) 1
(A) For given vectors and
the projection of the vector
the vector (B) If
and
(s) 0
(p) on
is are unit vectors satisfying
(q)
and
is equal to
(C) If , then k is (D) The value of c for which the vector LEVEL III MODEL QUESTIONS
(r) (s)
I. In a rhombus OABC, vector are respectively the position vectors of vertices A, B ,C with reference to O as origin . A point E is taken on the side BC which divides it in the ratio of 2:1 . Also, the line segment AE intersects the line bisecting the angle O intenally in point P. If CP, when extended meets AB in point F, then 1. The position vector of point P, is (A) (B) (C) (D) None of these 2. The position vector of point F , is (A)
(B)
(C)
3. The vector (A)
(B)
(C)
(D) None of these
, is given by (D) None of these
II A new operation * is defined between two non antiparallel vectors as
, where
is the angle between
4. The condition for which (B)
(C)
and
are perpendicular is
(A) 5.
is
(A)
(B) not defined (C) 0 (D) None of these
(D) None of these
6. For (A) (B) (C)
and .
= 0 is a necessary condition is a necessary condition is a sufficient condition, where
and
(D) None of these 7. Let
are non zero mutually perpendicular unit vectors. There is a
vector coplanar with (A) 8.
(B)
(C)
and
is
are unit vectors along the coordinate axes and
(B)
(C)
then q is
(D)
9. If the vectors only one real x then (A)
then minimum value of
(D)
makes angle q with (A)
and
and
are coplanar vectors for
is
(B) 2 (C) 4 (D)
10. If are unit vectors such that is equal to (A) 11. Let
(B)
(C)
then
(D) None of these
be three noncoplanar vectors and
the relations
are vectors defined by
, then the value of the expression
is equal to (A) 0 (B) 1 (C) 2 (D) 3 12. Let a,b,c be distinct nonnegative numbers. If vectors and lie in a plane then c is [IIT88] (A) the AM of a and b (B) the GM of a and b (C) the HM of a and b (D) equal to zero 13. Let equals
if
(C)
= is a unit vector such that
(A)
(B)
14. If
are three non coplanar vectors, then
, then
(D) equals
(A) 0 (B) 15. If
(C)
(D)
be three mutually perpendicular vectors of the same magnitude. If
a vector satisfies the equation given by (A)
(B)
16. If
,then is
(C)
(D)
are unit coplanar vectors, then the scalar triple
product
=
(A) 0 (B) 1 (C)
(D)
17. Let and . Then depends on [IIT2002] (A) only x (B) only y (C) Neither x Nor y (D) both x and y PRACTICE QUESTIONS 18. Let
and
. If
is a unit vector, then the maximum value
of the scalar triple product
is [IIT2003]
(A) (B) (C) (D) 19. The value of `a' so that the volume of parallelopiped formed by (A)
and (B) 3 (C)
20. If (A)
, (B)
becomes minimum is [IIT2004] (D) and
, then
is
(C) (D)
21. The unit vector which is orthogonal to the vector coplanar with the vectors (A) 22. If
(B)
(C)
and
and is
is [IIT2004]
(D)
are three non zero, non coplanar vectors and [IIT2005]
then the triplet of pairwise orthogonal vectors is (A)
(B)
(C)
(D)
23.
. A vector coplanar to
along
of magnitude
and
has a projection
, then the vector is [IIT2006]
(A) (B) (C) (D) None of these 24. The edges of a parallelepiped are of unit length and are parallel to noncoplanar unit vectors such that parallelepiped is [IIT2008] A)
. Then, the volume of the
(B)
(C)
(D)
25. Let two noncoplanar unit vector
and
form an acute angle. A point P
moves so that at any time t the position vector given by of
and
(A)
. Where P is farthest from origin O, let M be the length be the unit vector along and
. Then [IIT2008]
(B)
(C) 26. Let
. (where O is the origin ) is
(D) and
. If u is a unit vector, then for the maximum value
of the scalar triple product (A)
(B)
(C) (D) 27. For unit vectors b and c and any non zero vector a , the value of ` is
(A) (B) (C) (D) None of these 28. Three noncoplanar vectors a, b and c are drawn from a common initial point. The angle between the plane passing through the terminal points of these vectors and the vectors is (A)
(B)
29. If
(B) (D) None of these are non coplanar unit vectors such that
angle between (A)
(B)
and
(C)
then the
is :
(D)
30. The number of vectors of unit length perpendicular to vectors and is (A) 1 (B) 2 (C) 3 (D) 4 31. If is given that
and
,
and
If
, then (A)
(B)
(C)
(D)
32. If are non null vectors such that then (A) is equal to 1 (B) cannot be evaluated (C) is equal to zero (D)None of these 33. If are unit coplanar vectors, then scalar triple product is equal to (A) 0 (B) 1 (C) 34. Let
and
(D) are nonzero vectors such that
then is equal to (A) 0 (B) 1 (C) 2 (D)
,
Q. = i + 2J + 3k
= 3i + 2J + k and ( then p, q, r are Ans: 0, 10, –3. Q. If are unit vectors such that then the angles which makes with and are /3 & /2. Q. The vectors a, b, c are of same length taken pair wise, they form equal angles. If a = i+J b = J +k then C equals. (1, 0,1)
(1, 2, 3)
(–1, 1, 2)
(–1/3, 4/3, –1/3)
Q. Let a = 2l – J + k, b = l + 2J – k, c = i+ j 2k. A vector in the plane of b and c whose projection on a is of magnitude is A. 2i+ 3J – 3k
2i + 3J + 3k
2i+J+5k
2i+ J + 5k
Q. If a = (0, 1, 1), c =(1, 1, 1), then a vector be satisfying a b + c = 0 and a – b = 3 is Ans: (1, 1, 2) Q. If ( and at least one of the numbers , , is non zero. Then the vectors a, b, c are coplanar. Q. The scalars l and m are such that la + mb = c where a & b are monocollinear vectors then m=. Q. Let a, b, c be three vectors having magnitude 1, 1 &2, if a (ac) + b = 0 then the angle between a & c is /6. Q. The points A(–1, 3, 0) B(2, 2, 1) C(1, 1 ,3) determine a plane. The distance from the plane to the pt. D(5, 7, 8) is . Q. a b = c, b c = a, then
Q. P, Q, R be 3 mutually vector of same maq. If a vector x satisfies the eqn. + .then x = _____. Soln: p × [ x × p − q × p] + q × [ x × q − r × q ] + r × [ x × r − p × r ] = 0
k2{[
x − ( pˆ .x ) pˆ ) − (q − ( pˆ .q ) pˆ )] + [( x − (qˆ.x )qˆ ) − (r − ( qˆ.r )qˆ )]}(0) = 0
we know (
a .i )i + (a − j ) j + (a − k )k = a
parallelly – [( 3x – x = x=
pˆ .x ) pˆ + (qˆ.q )qˆ + ( rˆ.x )r ] = − x
p+q +r
(p + q + r) 2
Through the middle point M of the side AD of a parallelogram ABCD, the st. line BM is drawn cutting AC at R and CD produced at Q; prove that QR = 2 RB. 1.
2.
If
a = pi + sin θJ + k b= c
2i + PJ + k =i+J+k
a,b ,c
If + 1) /2. 3.
Let
are coplanar find all possible values of P& Q, P – 1, Q = (2n
a, b and d
be non coplanar vectors equally inclined to one
another at an angle . If
a ×b + b ×c
. (taking dot product with
=
a,b ,c,b × c
pa + qb + rc
).
. Find p, q, r in terms of
Ans: (p, q, r) =
1 − 2 cos θ 1 , , 1 + 2 cosθ 1 + 2 cos θ 1 + 2 cos θ
−1 2 cosθ −1 , , 1 + 2 cos θ 1 + 2 cos θ 1 + 2 cos θ
or
If one diagonal of a quadrilateral bisects the other then if also bisects the quadrilateral. 2. P, Q are the mid points of the nonparallel sides BC and AD of a trapezium ABCD. Show that APD = CQB. 1.
r
Find in terms of one parameter
3.
2
r=
r .n1 = 1 & r .n2 = 1
.[
2
n 2 − n1 .n2 n − n1 n 2 + 1 2 n1 (n1 × n2 ) (n1 × n 2 ) 2 n2
+t
n1 × n2
. r × a = b and r × c = d
Find the condition for the equations to be consistent assuming the condition for consistency to be satisfied, solve the eqns. 4.
b .d + a .d = 0 r =
[
1 ( d .b .a + d .a b + b 2 c ) [a b c ]
5. Prove that the projection of the line r=
q − a .n b .n . n + t b − .n a + n2 n 2
if [
ab c
] 0.
r = a + bt
normal
on the plane
r = a + b + pn
r.n
= q is
solving with the
plane. 6. In a tetra hedran OABC, OA BC, show that OB 2 + CA2 = OC2 + AB2. 7. Two medians of a are equal, show that the triangle is isoscelious.
1. If
A + B = a , A − a = 1, A × B = b , A=
a ×b + a a2
then
B=
1.
b × a + a (a 2 − 1) a2
Show that the eqn. of a line passing through the pt. with p. v.
equally in a lined to the vectors
a,b ,c
d
and
whose moduli are a, b, c is
a (b × c ) + b ( c × a ) c ( a × b ) r = d +α abc 2.
. A st. line ‘L’ cuts the lines AB, AC, AD of a gm ABCD at pts. B1, C1,
D1 respectively. If
AB1 = λ1 AB
,
AD1 = λ 2 AD, AC1 = λ3 AC
then prove that
1 1 1 = + λ3 λ1 λ 2
. 3. Consider a ABC, having AD as its median through the vertex A. let P be any pt. on its this median using vector methods prove that for every position of P on AD area of APB is equal to of APC. 4. ABCD is a gm and M be any pt. on the side AB. Using vector methods prove that for every position of M, AC & DM divide each other in the same ratio. Also locate the pt. ‘M’ such that AC and DM trisect each other. 5. If A1, A2,……..,An are n pts. on a circle one unit radius, prove that the sum of the squares of their mutual distances is not greater than n 2. 6. In a triangle ABC, D divides AC in the ratio 2:1. BD is produced to F such that DF = 2 BD. Prove that AF is to BC & equal to 2BC & CF is to median and is twice the median. 1.
Consider the vectors i + cos()J + cos()k, cos()i + J + cos()k and cos()i + cos()J + ak where . & are different angles. If these vectors are coplanar show that a is ind. of , , . (a = 1)
In a PQR, S and T are pts. on QR and PR respectively such that QS = 3SR and PT = 4 TR. Let M be the point of intersection of PS and QT. Determinethe ratio QM: MT (15:4). 2.
3.
Let
xˆ , yˆ
zˆ
and be unit vectors such that
xˆ + yˆ + zˆ = a , xˆ × ( yˆ × zˆ ) = b .( xˆ × zˆ ) × zˆ = c
terms of
a,b ,c
a .xˆ =
&
3 7 , a. yˆ = 2 4
,
a =2
, find
xˆ , yˆ , zˆ
in
.
1 xˆ = (3a + 4b + 8c ) 3 yˆ = −4c zˆ =
1.
4 (c − b ) 3
cos α
sin α
cos α
sin α
0
cos β cos γ
sin β 0 × cos β sin γ α − 1 cos γ
sin β sin γ
0 =0 1
0
The p. v. of pts. P & Q are 5i+7j2k and –3i+3j+ 6k respectively. The
vector
A=
3ij+k passes through the pt. P and the vector
2i+7J5k intersects vectors
1.f: R R, f(x) =
e x − e−x 2
A&B
B = −3i + 2 j + 4k
. Find p. .v of pts. of intersection.
11, on to, inverse graph.
2. Solve the eqn: f(x) = x3 – [x] = 5
Vector
,
Angle bisector AB 1.
In a triangle ABC the lengths of the to sides are known;
= 6,
AC = 8
and A = 90; AM and BN are bisectors of the angles A and B. Find the cosine of the angle between the vectors A (0) 6
N b 8
a
M b −c 10
B
C
a b AM = λ + 6 8 3b − 8a BN = µ 30 AM =
a2 b2 + = 2λ 36 64
BN = µ
4 5 5
, consider
1
Ans: C 6J A
10 M (…..) N 8i (…..)
B
AM .BN
AM
and
BN
.
1.
QUESTION BANK leveli The vector is 1) Null vector 2) unit vector 3) parallel to 4) a vector parallel to
2. If equal to
, 1)
then
is
2)
3) 4) 3. If A=(2,7,4) and AB=(2,5,3)then B= 1.(0,1,2) 2.(0,1,2) 3.(0,2,1,) 4.(0,2,1) 4. If and , then is true for 1) a = 1 2) a = 1 3) all real values of `a' 4) for no real values of `a' 5. Let A=2i+4k, B= , D =2i+k.Then the value of CD in terms of AB is 1. AB 2. AB 3. AB 4. AB 6. If the position vectors of the points A,B,C,D are (0,2,1),(3,1,1),(5,3,2),(2,4,1) respectively and if PA+PB+PC+PD = 0 then the position vector of Pis 1. 7.
2. and
3.
4.
are two vectors
where are two scalars. Then the lengths of the vector are equal for 1) all values of 2) only finite number of values of 3) infinite number of values of 4) No value of 8. If a is a vector so that its resultant with 3i+4j2k is i, then a = 1. 2i+4j+2k 2. 2i4j+2k 3. 2i4j2k4. 2i+4j2k 9. Let A,B,C be the vertices of a triangle.If AB=6i7j+k, AC=i6j+7kthen B C= 1. 7i+j+6k 2. 7ij+6k 3. 7ij6k 4. 7ij6k 10. If AB = 3i2j+k, BC = i+2jk, CD= 2i+j+3k OA= i+j+k,then the position vector of D is 1. 7i2j3k 2. 7i2j+3k 3. 7i+2j3k 4. 7i+2j+4k 11. Let A,B,C be the vertices of the triangle ABC and let then 1. 0 2.ai 3. 3a 4. (i+j+k) 12. If the position vectors of the vertices A,B,C are i+2j+3k, 3i4j+5k and 2i+3j+7k then the vector determined by the side BC is 1. 5i+7j+2k 2. 5i+7j+2k 3. 5i7j+2k4. 5i+7j2k 13. If the position vectors of P and Q arei+3j7k and 4i+5j+2k then PQ= 1. 3i+2j+9k 2. 3i2j+9k 3. 3i2j+9k 4.i+j+k 14. If a= 2i+4j5k, b=i+j+k, c=j+2k, then the unit vector parallel to a+b+c is
1. 2. 3. 4 . 15. If r1 = (3,2,1) r2 = (2,4,3) ,r3 = (1,1,2) then the modulus of the vector 2r1 3r2  4r3 is 1. 12 2. 13 3. 14 4. 15 16. The unit vector parallel to i3j5k is 1.
(i3j5k) 2. (i3j5k)
3. (i3j5k) 4. (i3j5k) 17. The unit vector parallel to the resultant vector 2i+3jk and 4i3j+2k is
1. 6i+k 2. 3.
4. 2i+k
18. The positon vectors of the points A,B,C are i+2jk , i+j+k , 2i+3j+2krespectively.If A is chosen as the origin then the position vectors of B and C are 1. i+2k, i+j+3k 2. j+2k, i+j+3k 3. j+2k, ij+3k 4. j+2k, i+j+3k 19. If AB = 2i3j+k ,CB = i+j+k,CD = 4i7j then AD= 1. 5i+11j+k 2. 5i11j 3. 5i+11j 4. 5i+11j 20. If A,B,C,D are the vertices of a quadrilateral taken in order thanAB+BC+CD+DA= 1. 4BD 2. 3BD 3. 0 4. 2BD 21. ABCDE is a pentagon.If forcesAB,AE,BC,DC,ED and AC act at a point then their resultant is 1. 3AC 2. AC 3. 2AC 4. 4AC 22. Let ABCD be a trapezium and let P,Q be the midpoints of the nonparallel sides AD,BC respectively.Then PQ = 1.AB+DC 2. (AB+BC) 3. (AB+DC) 4. (AB+BC) 23. Let A=(12,3),B=(3,1,5).C=(4,0,3).Then the triangle ABC is right angled at 1.
2.
3. 4. 24. The perimeter of the triangle whose vertices are the points 2ij+k, i3j5k, 3i4j4k is 1.
2.
3. 4. 25. Let A=2i+3j+4k,AB=5i +7j+6k. Then B= 1. (7i+10j+10k) 2. (7i10j+10k) 3. (7i+10j10k) 4. (7i+10j+10k) 26. If AO+OB=BO+OC then 1. A is the midpoint of BC 2. B is the midpoint of CA 3. C is the midpoint of AB 4. C divides AB in the ratio 1:2
27. The resultant of two concurrent forces nOP and mOQ is (m+n)OR.Then R divides PQ in the ratio 1. m : n 2. n : m 3. 1 : n 4. m : 1 28. Let A= 2i+7j, B= i+2j+4k, c = The ratio in which C divides AB internally is 1 1 : 4 2. 2 : 3 3. 3 : 2 4. 5 : 1 29. Let A =2i+3j+5k, B =7ik, If C divides AB in the ratio 1:2 then the position vector of C is 1. 3i+2j+k 2. 2i+3j+k 3. 3i+j+2k 4. i+2j+3k 30. If a+2b ,2a+b be the position vectors of the points A and B, then the position vector of the point C which divides AB internally in the ratio 2:1 is 1. 2. 3. 4. 31. 2ij+k,i3j+4 k be the position vectors of two points then the position vector which divides theabove two points in the ratio 2:3 is 1. 2. 3. 4. 32. The point C =(12/5, 1/5, 4/5) divides the line segment AB in the ratio 3:2 If B=(2,1,2) then A= 1. (3,1,1) 2. (3,1,1) 3. (3,1,1) 4. (3,1,1) 33. If b=3i4j and a are collinear and if amakes an obtuse angle with i and =10 then a = 1. 6i+8j 2. 6i+8j 3. 6i8j 4. 6i8j 34. Let a = (1,1,1), b =(5,3,3), c =(3,1,2) .The vectors which are collinear with c and whose lengths are equal to that of a+b are 1. (6,2,4),(6,2,4) 2. (6,2,4),(6,2,4) 3. (6,2,4),(6,2,4) 4. (6,2,4),(6,2,4) 35 Let A and B be points with position vectors If the point `C' on OA is such that then AD is 1.
2.
3.
4.
with respect to origin O.
is parllel to
and
36. P,Q,R,S have position vectors .Then 1. PQ and RS bisect each other 2. PQ and PR bisect each other 3. PQ and RS trisect each other 4. QSand PR trisect each other
respectively such that
37. The position vectors of the points A,B,C are respectively divides then
in the ratio 3:4 and Q divides
If P
in the ratio 2:1 both externally
= 1.
2.
3.
4.
38. If are the position vectors of A and B then one of the following points lie on 1.
2.
3. 4. RATIO 39. Let A =(3,4,8) ,B=(5,6,4). Then the ratio in which the XOY plane divides the line segment AB is 1. 2 : 1 2. 1 : 2 3. 3 : 5 4. 2 : 3 40. Let A =(3,4,8) ,B=(5,6,4). Then the coordinates of the point in which the XY plane or XOY plane divides the line segment AB is 1.(7,8,0)
2.
3. 4. 41. Let A =i+2j+3k , B= 4i+2j, C=2i+2j+2k. Then the ratio in which C divides AB is 1. 3 : 4 2. 1 : 3 3. 1 : 2 4. 1 : 1 42. If 3a+4b7c =0 then the ratio in which C(c) divides the join of A(a) and B(b) is 1. 1 : 2 2. 2 : 3 3. 3 : 2 4. 4 : 3
43. If a,b,c,d are the position vectors of the points A ,B,C,D respectively such that 3a+5b3c5d =0 then AB intersectsCD in the ratio 1. 2 : 3 2. 3 : 2 3. 3 : 5 4. 5 : 3 DIRECTION COSINES & RATIOS 44. A straight line is inclined to the axes of Y and Z at angles respectively. The inclination of the line with the Xaxis is 1. 2. 3. 4. 45. The direction cosines of i+2j+2k are 1. (1,2,2) 2.
3.
46. A line makes an angle .Then 1. 1
2.2
4. with the coordinate axes
= 3. 3/2
4. 1/3
47. A line makes an angle with the X,Y,Z axes .Then = 1. 1 2. 2 3. 3/2 4. 4 48. The direction cosines of two lines are l1, m1, n1 ; l2, m2, n2 .Then the value of = 1.1 2. 0 3. 4 4. 2 49. If e = li+mj +nk is unit vector ,the maximum value of lm+mn+nl is 1.
2. 0
3. 1
4.
50. If l.m.n are the d.c's of a vector if ,then the maximum value of lmn is 1. ¼ 2. 3/8 3. 1/2 4. 3/16 51. If OA =3i+jk,AB =
and AB has the direction ratios 1,1,2 then OB =
1. 2. 3. 4. CENTROID 52. Let G and G  be the centroids of the triangles ABC and A B C  respectively . Then AA+BB+CC = 1. 2GG 2.3GG 3. 3GG 4. 3/2 GG 53. Let A= 2i+4jk ; B = 4i+5j+k .If the centroid G of the triangle ABC is 3i+5jk then the position vector of C is 1. 3i6j+3k 2. 3i6j3k 3. 3i6j+2k 4.3i+6j3k
54. If the position vectors of A,B,C are 3i+jk,i2j, 2ij+3k then the position vector of the centroid of the triangle ABC is 1.2i2/3j+2/3k 2.2i+2/3j+2/3k 3.3i2j+2k 4. 2i2/3j2/3 55. If G is the centroid of the triangle ABC then GA+GB+GC = 1. AB 2.BC 3.4GA 4.0 56. If and be the vertices of a triangle whose circumcentre is the origin then orthocentre is given by 1)
2)
3)
4)
57. Let A = i+6j+6k , B = 4i+9j+6k, .If G is the centroid of the triangle ABC is 1. A right angled triangle 2. A right angled isosceles triangle 3. An isosceles triangle 4. An equilateral triangle 58. The triangle ABC is defined by the vertices A = (0,7,10) B = (1,6,6) and C = (4,9,6) let D be the foot of the attitude from B to the side AC . then BD is 1) 2) 3) 4) 59. Let A(a), B(b),C(c) be the vertices of the triangle ABC and let D,E,F be the mid points of the sides BC,CA,AB respectively.If P divides the median AD in the ratio 2:1 then the position vector of P is 1.0 2.a+b+c 3. 4. 60. The midpoints D,E,F of the sides BC,CA,AB of vectors 1.
, 2.
Then the centroid of 3.
have position
is the point :
4.
61. If G is the centroid of triangle ABC, then = 1.1/4 2.1/3 3.2/3 4.4/9 TRIANGLE 62. If D is the mid point of the side BC of triangle ABC, then AB+AC=
1. AD 2.1/2AD 3.2AD 4.4AD 63. Let ABC be a triangle and AD,BE,CF be its medians then AD+BE+CF= 1.4AB 2.3BC 3.4CA 4.0 64. Let ABC be a triangle and let D,E,F be the midpoints of the sides BC,CA,AB respectively. Then
=
1. 2. 3. 4. 65. Let ABC be a triangle and let D,E be the midpoints of the sides AB,AC respectively, then BE+DC= 1. BC 2. 3. 4. 66. Let ABC be a triangle and let S be its circumcentre and O be its orthocentre. The SA +SB+SC= 1. 4SO 2. 3SO 3. 2SO 4.SO 67. Let ABC be a triangle and let S be its circumcentre and O be its orthocentre. The OA +OB+OC= 1. O 2. 2SO 3. 2OS 4. 3OS  68. If O is the circumcentre and O is the orthocentre of a triangle ABC and if AP is the circumdiameter then AO+OB+OC= 1. OA 2.O'A 3.AP 4.AO 69. If AB = 3i+4k and BC = the length of the median AM is
are the sides of the triangle ABC, then
1. 2. 3. 4. 70. If D,E,F the midpoints to the sides of a triangle ABC then area of 1. 71. Let
2.
3.
4.
be the distinct real numbers. The points with position vectors are
1.collinear 2.form an isosceles triangle 3.right angled triangle 4.equilatetral triangle 72. If the vectors
=
form a triangle then = 1.6 2.6 3.12 4.1 73. The incentre of the triangle formed by the points i+j+k , 4i+j+k , 4i+5j+k is 1. 2.i+2j+3k 3.3i+2j+k 4.i+j+k 74. The position vectors of A,B,C are vector of the circumcentre of the triangle ABC is 1.
2.
3.
4.
Then the position
COLLINEAR 75. If and are the vectors whose directions are neither parallel non coincident, then 1)
0,
76. If
and
,
and
0 2]
implies 1,
1 3]
1,
1 4]
1,
1
are two noncollinear vectors, then the points are collinear if
1) 2) 3) 4) 77. Let a, b be two noncollinear vectors .If A = (x+4y)a +(2x+y+1)b, B=(y2x+2)a + (2x3y1)b and 3A=2B ,then (x,y)= 1.(1,2) 2.(1,2) 3.(2,1) 4.(2,1) 78. Let a, b be two noncollinear vectors .If c = (x2) a+b , d=(2x+1) ab are collinear then x= 1. 1 2. 1/2 3. 1/3 4. 1/4 79. If i+pj+k , 2i+3j+qk are like parallel vectors then (p,q)= 1. 2.(2,2) 3. 4. 80. If the points A(a),B(b),C(c) satify the relation 3a8b+5c=0 then the points are 1.vertices of an equilateral triangle 2.collinear
3.vertices of a right angled triangle 4.vertices of a isosceles triangle 81. If the vectors 2i+aj+k , 5i+3j+bkare collinear , then (a,b)= 1. (6/5,5/2) 2.(6/5,5/2) 3.(12,25) 4.(12,25) 82. If the vectors 2i3j+6k , b are collinear
=14 , then b=
1. 2. 3. 4. 83 If 4i+5jk , 8i+aj2k are parallel vectors then a = 1.10 2.5 3.10 4.5 84. If a, a are collinear and are in the same direction then is 1.=0 2.>0 3.
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