Solve $p \times h = p \times q$

Question

I'm trying to solve $p \times h = p \times q$ for $h$, given $p \ne 0$. From the equation, it seems to me that $h$ has to be in the plane of $p$ and $q$ which would then imply $h \cdot (p \times q)=0 $ but that doesn't help me solve the equation. I can't seem to get anywhere using $|p||h|\sin \theta_1 = |p||q|\sin \theta_2$ either.

Answer 1

We have $$|p\times h|^2=|p|^2|h|^2-(p\cdot h)^2$$$$|p\times q|^2=|p|^2|q|^2-(p\cdot q)^2$$ Let $p=(p_1,\cdots,p_n)$, $q=(q_1,\cdots,q_n)$ and $h=(h_1,\cdots,h_n)$. Then $$\sum_{i=1}^np_i^2\sum_{i=1}^nh_i^2-\left(\sum_{i=1}^n p_ih_i\right)^2=\sum_{i=1}^np_i^2\sum_{i=1}^nq_i^2-\left(\sum_{i=1}^n p_iq_i\right)^2$$ from which you can hopefully solve for $h$ for small $n$.