Question

$x$ component of $\overline{a}$ is twice of its $y$-component. If the magnitude of the vector is $5\sqrt{2}$ and it makes an angle of ${135}^{\circ }$ with $z$-axis then the vector is

• A. $2\sqrt{3},\sqrt{3},-3$
• B. $2\sqrt{6},\sqrt{6},-6$
• C. $2\sqrt{5},\sqrt{5},-5$
• D. $\sqrt{6},\sqrt{6},-6$

Answer 1

Answer: (C)

$2\sqrt{5},\sqrt{5},-5$

$x_{component}=5\sqrt{2}cos\alpha$
$y=5\sqrt{2}cos\beta$
$z=5\sqrt{2}cos\gamma =5\sqrt{2}cos\left({135}^0\right)$
$=-5$
Now
$x=2y$
O
$5\sqrt{2}cos\alpha =2\left(5\sqrt{2}\right)cos\beta$
$cos\alpha =2cos\beta$
Now
$co{s}^2\alpha +co{s}^2\beta +co{s}^2\gamma =1$
or
$4co{s}^2\beta +co{s}^2\beta +\frac{1}{2}=1$
$5co{s}^2\beta =\frac{1}{2}$
$cos\beta =\frac{1}{\sqrt{10}}$
Hence
$x=5\sqrt{2}cos\alpha$
$=5\sqrt{2}\left(\frac{2}{\sqrt{10}}\right)$
$=2\sqrt{5}$.
Therefore
$y=\frac{x}{2}$
$=\sqrt{5}$
Hence
$(x,y,z)=(2\sqrt{5},\sqrt{5},-5)$