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

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### 1 Answer

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.**