Question

$P$ and $Q$ are the points $(-1,2,1)$ and $(4,3,5)$ respectively. Find the projection of $PQ$ on a line which makes angles of ${120}^o$ and ${135}^o$ with $y$ and $z$ axes respectively and an acute with $x-axis$ ?

Answer 1

$\alpha ,\beta ,\gamma$ be the angles made by the coordinate axes.

$\therefore \cos {}^2\alpha +\cos {}^2\beta +\cos {}^2\gamma =1$

$\Rightarrow \cos {}^2\alpha +\cos {}^{2}120+\cos {}^{2}135=1$ (given $\beta ={120}^{\circ },\gamma ={135}^{\circ }$)

$\Rightarrow \alpha ={60}^{\circ }$

$\therefore$ Directions of the projection $(\cos \alpha ,\cos \beta ,\cos \gamma )\equiv (\frac{1}{2},\frac{-1}{2},\frac{-1}{\sqrt{2}})$

$\therefore \overrightarrow{PQ}=5\hat{i}+\hat{j}-4\hat{k}$

projection of $\overrightarrow{PQ}$ on line with directions $(\frac{1}{2},\frac{-1}{2},\frac{-1}{\sqrt{2}})$ will be .

$\Rightarrow \overrightarrow{PQ}.(\frac{1}{2}\hat{i}-\frac{1}{2}\hat{j}-\frac{1}{\sqrt{2}}\hat{k})=(5\hat{i}+\hat{j}-4\hat{k}).(\frac{1}{2}\hat{i}-\frac{1}{2}\hat{j}-\frac{1}{\sqrt{2}}\hat{k})=2+2\sqrt{2}$

In vector form, $(2+2\sqrt{2})(\frac{1}{2}\hat{i}-\frac{1}{2}\hat{j}-\frac{1}{\sqrt{2}}\hat{k})$