Dot product or scalar product of vectors

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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 _________

Moderate

Solution

$\begin{array}{l} \vec{A B} = (\hat{i}) - (3 \hat{j}) + 8 \hat{k} \\ \vec{C D} = 4 \hat{i} - 4 \hat{j} + 7 \hat{k} \end{array}$
$Projection of \vec{A B} on \vec{C D} will be \frac{\vec{A B} \cdot \vec{C D}}{| \vec{C D} |} = \frac{4 + 12 + 56}{\sqrt{16 + 16 + 49}} = \frac{72}{9} = 8$

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