Question

$P(1,0,1),Q(2,0,-3),R(-1,2,0)$ and $S(3,-2,-1)$, then find the projection length of $\overrightarrow{PQ}$ on $\overrightarrow{RS}$

Answer 1

Consider the given vector points,

$P(1,01),Q(2,0-3),R(-1,2,0)$ and $S(3,-2,-1)$

Now,

$\overrightarrow{PQ}=\hat{i}+0j-4\hat{k}$ and $\overrightarrow{RS}=4\hat{i}-4\hat{j}-\hat{k}$

Now projection of $\overrightarrow{PQ}$ on $\overrightarrow{RS}$ =

$\frac{\overrightarrow{PQ.}\overrightarrow{RS}}{\left|\overrightarrow{RS}\right|}=\frac{(\hat{i}+0j-4\hat{k}).(4\hat{i}-4\hat{j}-\hat{k})}{|4\hat{i}-4\hat{j}-\hat{k}|}$

$=\frac{1+0+4}{\sqrt{4^2+{(-4)}^2+{(-1)}^2}}=\frac{5}{\sqrt{33}}$

Hence, this is the answer.