What is the length of the projection of $3 \hat{\mathrm{i}}+4 \hat{\mathrm{j}}+5 \hat{\mathrm{k}}$ on the…
What is the length of the projection of $3 \hat{\mathrm{i}}+4 \hat{\mathrm{j}}+5 \hat{\mathrm{k}}$ on the $\mathrm{xy}$-plane ?
3
5
7
9
Solution
$x y$-plane is perpendicular to $z$ - axis. Let the vector
$\overrightarrow{\mathrm{a}}=3 \mathrm{i}+4 \mathrm{j}+5 \mathrm{k}$ make angle $\theta$ with $\mathrm{z}$ - axis, then it makes $90-\theta$ with $x y$-plane. unit vector along $\mathrm{z}$-axis is $\mathrm{k}$.
So, $\cos \theta=\frac{\overrightarrow{\mathrm{a}} \cdot \hat{\mathrm{k}}}{|\overrightarrow{\mathrm{a}}| \cdot|\hat{\mathrm{k}}|}=\frac{(3 \mathrm{i}+4 \mathrm{j}+5 \mathrm{k}) \cdot \mathrm{k}}{|3 \mathrm{i}+4 \mathrm{j}+5 \mathrm{k}|}$
$=\frac{5}{5 \sqrt{2}}=\frac{1}{\sqrt{2}} \Rightarrow \theta=\frac{\pi}{4}$
Hence angle with $x y-$ plane $\frac{\pi}{2}-\frac{\pi}{4}=\frac{\pi}{4}$
projection of $\vec{a}$ on xy plane $=|\vec{a}| \cdot \cos \frac{\pi}{4}$ $=5 \sqrt{2} \times \frac{1}{\sqrt{2}}=5$