IA-04Scalar (Dot) Product of Two Vectors(23-27)

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4. SCALAR (DOT) PRODUCT OF TWO VECTORS

2.

The scalar product or (dot product) : The scalar product of two vectors a and b is a scalar and is defined as the product of the magnitude of a and b and the cosine of the angle θ between them. i.e., a . b =| a | . | b | .cosθ , 0 ≤ θ < π . Two non zero vectors a and b are perpendicular if and only if their dot product is zero.

3.

Projection of a and b is

1.

| a.b | (a . b )b the orthogonal projection of a and b is or ( a .bˆ)bˆ . |b| | b |2

The component vector of a along b is

(a. b ) b | b |2

or ( a .bˆ)bˆ .

The component vector of a perpendicular to b is a −

4.

(a . b ) b | b |2

.

Note : The component vector of a along b is equal to the orthogonal projection of a on b . If a = a1i + a 2 j + a 3 k , b = b1i + b 2 j + b 3 k , then i) a . b = a1b1 + a 2 b 2 + a 3 b 3 ii) If θ is the angle between a and b , then a.b | a || b |

iii)

=

cosθ =

a 1b1 + a 2 b 2 + a 3 b 3

∑a . ∑b (a b − a b ) sinθ = ∑ (∑ a )(∑ b ) 2 1

2 1

2 3 2 1

2

3 2 2 1

iv) a . b = b . a (dot product is commutative) v) a .(- b ) = ( − a ).b = -( a . b ) λ a .μ b = λμ( a . b ) = μ a .λ b vi) a . a =| a | 2 = a 2 vii) viii) ( a ± b ) 2 =| a | 2 + | b | 2 ±2 a . b ix) ( a + b ) 2 = ( a − b ) 2 = 2( a 2 + b 2 ) and ( a + b ) 2 − ( a − b ) 2 = 4 a.b

x) ( a + b + c ) 2 = a 2 + b 2 + c 2 + 2( a.b ) + 2( b.c ) + 2( c.a )

5.

xi) a 2 − b 2 = ( a + b ).( a − b ) xii) a.( b + c ) = a.b + a.c , dot product is left distributive over vector addition. xiii) ( a + b ).c = a.c + b.c , dot product is right distributive over vector addition. For the unit orthognormal vectors i, j, k i) i.i = j. j = k . k = 1 ii) i. j = j.k = k . i = 0

6.

For any vector r, r = (r.i )i + (r. j) j + (r.k )k

7. 8.

Work done by a force F in displacing a particle from A to B is given by W = F.AB If the length of the perpendicular from the origin be p ( > 0) and nˆ be the unit vector perpendicular to the plane then the equation of the plane is r.nˆ = p . 1

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Scalar (dot) Product of Vectors 9.

The vector equation of a plane through a point a and perpendicular to a unit vector nˆ is (r − a ).nˆ = 0 .

10. If π1 and π2 be two planes whose equations are r1.m1 = q1, r2 .m2 = q 2 , then the angle between the ⎛

m .m

⎞

1 2 ⎟⎟ . planes is Cos −1 ⎜⎜ ⎝ | m1 | . | m 2 | ⎠

11. Perpendicular distance from the origin to the plane (r − a ).nˆ = 0 is a .nˆ , where a is the position vector of a point in the plane and nˆ is the unit vector perpendicular to the plane.

2

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