Prove that the cross product of two parallel vectors is 0.

Two vectors are parallel if the ratio of the coefficients
of their components have the same ratio. If A = ai + bj + ck, B is parallel to A if B =
r*A = rai + rbj + rck.

For two vectors A and B where A =
a1*i + b1*j + c1*k and B = a2*i + b2*j + c2*k, the cross product AxB is given by (b1*c2
- c1*b2)i - (a1*c2 - c1*a2)j + (a1*b2 - b*a2)k

As A is
parallel to B

AxB = (b*rc - c*rb)i - (a*rc - c*ra)j + (a*rb
- b*ra)k

=> (rbc - rbc)i - (rac - rac)j + (rab -
rab)k

=>
0

This proves that the cross product of
parallel vectors is zero.