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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