Mechanics II SS2 Further Mathematics Lesson Note

VECTORS OR CROSS PRODUCT ON TWO OR THREE DIMENSION , CROSS PRODUCT OF TWO VECTORS AND APPLICATION OF CROSS PRODUCT 

Vector Product Of Two Vectors 

Given two vectors and whose directions are inclined at an angle0, their vector product is defined as a vector ‘r’ whose magnitude is absin0 and whose directions is perpendicular to both a and b also being positive relative to a rotation from the vector a and b also being positive relative to a rotation from the vector a to the vector b.

The vector product of a and b is designated 

a×b

Thus: 

r = a x b =|a| |b| sin0.Ü where Ü is a unit vector perpendicular to the plane of a and b . 

Properties of vector Product 

 x = |b||a| sin(-) 0 <<

= – |a||b| sin (-0) Ü o≤ 0 ≤ π

= -a × b

Thus the vector product of two vectors is not commutative .

(ka) x b = ax (kb )

          = k (axb)

= k |a|b| sin0Ü )

Where k is a scalar.

ax (b + c)     = ax b + ax c

ASSIGNMENT 

1. Given that  p = 2i + 3j +4k  and  q= 5i – 6j +7k. Find:
2. p x q  
3. (p + q ) . ( p-q)

2i. Given that  a = 4i – 5j + 2k  and  b =  -7i + 3j – 6k  find the scalar product of a and b  

1. Find the direction  cosine  2a + 3b

3. Find the angle between p = 6i + 2j – 4k and q = 9i + 5j