Question

A unit vector a makes an angle $\frac{\pi }{4}$ with $Z$-axis, if $\hat{a}+\hat{i}+\hat{j}$ is a unit vector, then $a$ is equal to

• A. $\frac{\hat{i}}{2}+\frac{\hat{j}}{2}+\frac{\hat{k}}{2}$
• B. $\frac{\hat{i}}{2}+\frac{\hat{j}}{2}-\frac{\hat{k}}{\sqrt{2}}$
• C. $-\frac{\hat{i}}{2}-\frac{\hat{j}}{2}+\frac{\hat{k}}{\sqrt{2}}$
• D. $\frac{\hat{i}}{2}-\frac{\hat{j}}{2}-\frac{\hat{k}}{\sqrt{2}}$

Answer 1

Answer: (C)

Let $a=\widehat{li}+m\hat{j}+n\hat{k}$ makes an angle $\frac{\pi }{4}$ with Z-axis
Also, $l^2+m^2+n^2=1$
Here, $n=\cos \frac{\pi }{4}=\frac{1}{\sqrt{2}},l^2+m^2=\frac{1}{2}....$(i)
$\therefore a\hat{a}=1\hat{i}+m\hat{j}+\frac{\hat{k}}{\sqrt{2}}$
$\Rightarrow a+\hat{i}+\hat{j}=(l+1)\hat{i}+(m+1)\hat{j}+\frac{\hat{k}}{\sqrt{2}}$
$\Rightarrow la+\hat{i}+\hat{j}l=\sqrt{(l+1{)}^2+(m+1{)}^2+{\left(\frac{1}{\sqrt{2}}\right)}^2}$
$\Rightarrow 1=l^2+m^2+2+2l+2m+\frac{1}{2}$
$\Rightarrow l+m=-1$ [From Eq.(i)]
$\Rightarrow l^2+m^2+2lm=1$
$\Rightarrow 2lm=\frac{1}{2}\Rightarrow l=m=-\frac{1}{2}$
$(\because l=m=\frac{1}{2}$ is not satisfied the given equation)
$\therefore a=-\frac{\hat{i}}{2}-\frac{\hat{j}}{2}+\frac{\hat{k}}{\sqrt{2}}$