The projection of the line segment joining

Question:

The projection of the line segment joining the points $(1,-1,3)$ and $(2,-4,11)$ on the line joining the points $(-1,2,3)$ and $(3,-2,10)$ is______.

Solution:

Let $P(1,-1,3), Q(2,-4,11), R(-1,2,3)$

and $S(3,-2,10)$

Then, $\overrightarrow{P Q}=\hat{i}-3 \hat{j}+8 \hat{k}$

$\overrightarrow{R S}=4 \hat{i}-4 \hat{j}+7 \hat{k}$

Projection of $\overrightarrow{P Q}$ on $\overrightarrow{R S}$

$=\frac{\overrightarrow{P Q} \cdot \overrightarrow{R S}}{|\overrightarrow{R S}|}=\frac{4+12+56}{\sqrt{(4)^{2}+(4)^{2}+(7)^{2}}}=8$

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