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5. VECTOR PRODUCT OF TWO VECTORS Synopsis: 1.

The cross product of two vectors a and b is a vector and is defined as | a | . | b | sinθ.nˆ , where nˆ is the unit vector perpendicular to the plane containing a and b such that a , b , nˆ form a right handed system.

2.

Let F be a force directed along a line. Let O be a point (origin). Let OP = r be the position vector of any point P on the line of action of F . Then rxF gives the moment of the force F about the point O.

3.

a x b = O ⇔ either a = O (or) b = O (or) a is parallel to b .

4.

i j If a = a1i + a 2 j + a 3 k , b = b1i + b 2 j + b 3 k , then a x b = a1 a 2 b1 b 2

5.

If a x b ≠ b x a but a x b = - b x a (i.e., cross product is anti commutative)

6.

If i , j, k is a orthonormal triad of unit vectors forming a right handed system then i xj = -jx i ; jxk = - k xj = i ; k x i = - i xk = j and i x i = j xj = k xk = 0 .

7.

Unit vector perpendicular to the both the vectors a and b is ±

8.

If a and b are the adjacent sides of a parallelogram then vector area of the parallelogram is and area = | a x b | sq.units.

9.

If a and b are two adjacent sides of a triangle then area of the triangle is

k a3 b3

axb | axb |

10. If a , b , c are the vertices of a triangle then the area of the triangle is

.

1 | axb | 2

a xb

sq.units.

1 | b xc + cxa + a x b | sq.units. 2

11. The points a , b , c are collinear if a x b + b xc + cx a = 0 . 12. If a , b , c are the position vectors of the points A, B, C then the perpendicular distance from C to

the straight line AB is

| b xc + c x a + a x b | |b−a|

or

| ACx AB |

.

| AB |

13. If l1, m1, n1 and l2, m2, n2 are the direction ratios of a straight line then cosθ = 14. For every vector i) a xa = 0 ii) | a x b |=| b x a |

iii) a x(b + c) = a xb + a xc cross product is left distributive over vector addition iv) ( a + b )xc = a xc + b xc cross product is right distributive over vector addition 15. For any two vectors a and b , ( a x b ) 2 + ( a.b ) 2 =| a |2 | b |2

1

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(l1l 2 + m1m 2 + n1n 2 ) ∑ l12 ∑ l 22

.

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5. VECTOR PRODUCT OF TWO VECTORS Synopsis: 1.

The cross product of two vectors a and b is a vector and is defined as | a | . | b | sinθ.nˆ , where nˆ is the unit vector perpendicular to the plane containing a and b such that a , b , nˆ form a right handed system.

2.

Let F be a force directed along a line. Let O be a point (origin). Let OP = r be the position vector of any point P on the line of action of F . Then rxF gives the moment of the force F about the point O.

3.

a x b = O ⇔ either a = O (or) b = O (or) a is parallel to b .

4.

i j If a = a1i + a 2 j + a 3 k , b = b1i + b 2 j + b 3 k , then a x b = a1 a 2 b1 b 2

5.

If a x b ≠ b x a but a x b = - b x a (i.e., cross product is anti commutative)

6.

If i , j, k is a orthonormal triad of unit vectors forming a right handed system then i xj = -jx i ; jxk = - k xj = i ; k x i = - i xk = j and i x i = j xj = k xk = 0 .

7.

Unit vector perpendicular to the both the vectors a and b is ±

8.

If a and b are the adjacent sides of a parallelogram then vector area of the parallelogram is and area = | a x b | sq.units.

9.

If a and b are two adjacent sides of a triangle then area of the triangle is

k a3 b3

axb | axb |

10. If a , b , c are the vertices of a triangle then the area of the triangle is

.

1 | axb | 2

a xb

sq.units.

1 | b xc + cxa + a x b | sq.units. 2

11. The points a , b , c are collinear if a x b + b xc + cx a = 0 . 12. If a , b , c are the position vectors of the points A, B, C then the perpendicular distance from C to

the straight line AB is

| b xc + c x a + a x b | |b−a|

or

| ACx AB |

.

| AB |

13. If l1, m1, n1 and l2, m2, n2 are the direction ratios of a straight line then cosθ = 14. For every vector i) a xa = 0 ii) | a x b |=| b x a |

iii) a x(b + c) = a xb + a xc cross product is left distributive over vector addition iv) ( a + b )xc = a xc + b xc cross product is right distributive over vector addition 15. For any two vectors a and b , ( a x b ) 2 + ( a.b ) 2 =| a |2 | b |2

1

Powered by www.myengg.com

(l1l 2 + m1m 2 + n1n 2 ) ∑ l12 ∑ l 22

.

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