For the vectors p= (1, -2, 3), q= (2, 1, 3), and r= (1, 1, 0). Show the following to be true.

Question

For the vectors p= (1, -2, 3), q= (2, 1, 3), and r= (1, 1, 0). Show the following to be true.

1. a) The vector (p x q) x r can be written as a linear combination of p and q
2. b) (p x q) x r = (p · r) q - (q · r)p

Answer 1

If you do b) first, you have already done a). Of course, a is also easily understood via understanding cross product.

b) pxq = the determinant of

i j k

1 -2 3

2 1 3

=

-9i + 3j + 5k

Then, (pxq) x r = the determinant of

i j k

-9 3 5

1 1 0

=

-5i + 5j - 12k

(p.r)q =(q.r)p =

(1, -2, 3).(1, 1, 0) (2, 1, 3) - (2, 1, 3).(1, 1, 0) (1, -2, 3) =

-(2, 1, 3) -3(1, -2, 3) =

-(2, 1, 3) + (-3, 6, -9) =

(-5, 5, -12)

This equals (pxq) x r above

a) (pxq)xr) = -p -3q (see b)