Question

If $\overrightarrow{P}=\hat{i}+2\hat{k}$ and $\overrightarrow{Q}=2\hat{i}+\hat{j}-2\hat{k}$ are two vectors, find the unit vector parallel to $\overrightarrow{P}\times \overrightarrow{Q}$. Also find the vector perpendicular to $P$ and $Q$ of magnitude $6$ units.

Answer 1

Let R be the vector perpendicular to P and Q

$\overrightarrow{R}$ = $\overrightarrow{P}\times \overrightarrow{Q}$

= $(\hat{i}+2\hat{k})\times (2\hat{i}+\hat{j}-2\hat{k})$

$\overrightarrow{R}=-2\hat{i}+6\hat{j}+\hat{K}$

so unit vector along $(\overrightarrow{P}\times \overrightarrow{Q})$

And S is a vector along R vector having magnitude 6

$\overrightarrow{S}=6\times \frac{\overrightarrow{P}\times \overrightarrow{Q}}{|\overrightarrow{P}\times \overrightarrow{Q}|}$

= $6\times \frac{-2\hat{i}+6\hat{j}+\hat{k}}{\sqrt{41}}$