Visualizing planes, cross-products and lines ; in this question, how is w x n the direction of PQ?

Question

I'd like some geometrical insight if possible. Why is PQ in the same direction as the cross product of w and n? PQ lies in the plane, so it should be perpendicular to n. How is it also perpendicular to w? Diagrams will be hugely appreciated.

Answer 1

You have $$\overrightarrow{AQ}\parallel\underline{n}$$

By definition of cross-product, $$\underline{w}\perp\underline{n}$$ and $$\underline{w}\perp\overrightarrow{AP}$$

Therefore $\underline{w}$ is normal to the plane $APQ$

Thus $\underline{n}\times\underline{w}$ is parallel to plane $\Pi$ and to plane $APQ$ and is therefore parallel to $\overrightarrow{PQ}$ which is the line of intersection of these two planes.

So the equation of the line $PQ$ is $$\underline{r}=\overrightarrow{OP}+\lambda\underline{n}\times\underline{w}$$