If $p \times q = 3p \times r$, then $q - 3r = \lambda p$

Question

Three non-zero and non-parallel vectors $p,q$ and $r$ are such that $p \times q = 3p \times r$. Show that $q - 3r = \lambda p$, where $\lambda$ is a scalar.

My attempt: We have

$$p \times q = 3p \times r $$ $$\Rightarrow p \times q - 3p \times r=0$$ $$p \times q -p\times r - 2p\times r=0$$

Then I stuck here. Any hint would be much appreciated.

Answer 1

Consider $ p \times q = 3p \times r \implies p \times q = p \times 3r $

Now , $ p \times q - p \times 3r = 0 \implies p \times (q-3r) = 0 $

Now if cross product of two vectors is 0 , one must a multiple of other so that cross-product of the same vectors is 0.

So , $ q-3r = \lambda p $ which implies on substituting $ p \times \lambda p = \lambda p \times p = \lambda . 0 = 0 $

Thus proved