Question

Find a vector $\overrightarrow{a}$ of magnitude $5\sqrt{2}$ making an angle of $\frac{\pi }{4}$ with x-axis, $\frac{\pi }{2}$ with $y-$axis and an acute angle $\theta$ with $z-$axis.

Answer 1

Given, vector $|V|=5\sqrt{2}$

$\Rightarrow a^2+b^2+c^2=50$

The direction of cosines of $\overrightarrow{A}$ are

Angle with x - axis $cos\alpha =\frac{x}{|\overrightarrow{A}|}=\frac{x}{5\sqrt{2}}$

$cos\alpha =\frac{\pi }{4}=\frac{1}{\sqrt{2}}$

So, $\frac{1}{\sqrt{2}}=\frac{x}{5\sqrt{2}}$

$x=5$

Angle with y - axis

Similarly for $cos\alpha =\frac{y}{|\overrightarrow{A}|}=\frac{y}{5\sqrt{2}}$

$cos\alpha =\frac{\pi }{2}=0$

So, $0=\frac{y}{5\sqrt{2}}$

$y=0$

$a^2+b^2+c^2=50$

$25+0+c^2=50$

$c^2=25$

$c=\pm 5$

Angle with z-axis is acute.

$cos\theta >0$ so $c=5$

Therefore, the vector $V=5\overrightarrow{i}+0\overrightarrow{j}+5\overrightarrow{k}.$