Question

Find a unit vector $\overrightarrow{a}$ which makes an angle $\left(\pi /4\right)$ with axis of z & is such that $\overrightarrow{a}+\hat{i}+\hat{j}$ is a unit vector

• A. $\frac{-1}{2}i-\frac{1}{2}j+\frac{1}{\sqrt{2}}k$
• B. $-i-j+\sqrt{2}k$
• C. $\frac{1}{2}i-\frac{1}{2}j-\frac{1}{\sqrt{2}}k$
• D. $i-j-\sqrt{2}k$

Answer 1

The correct option is A $\frac{-1}{2}i-\frac{1}{2}j+\frac{1}{\sqrt{2}}k$

Let the required unit vector be $\overrightarrow{a}=x\hat{i}+y\hat{j}+z\hat{k}$
$\overrightarrow{a}$ makes angle $\frac{\pi }{4}$ with $Z$ axis.
$\Rightarrow z=\cos \frac{\pi }{4}=\frac{1}{\sqrt{2}}$ ----(1)
$\overrightarrow{a}$ and $\overrightarrow{a}+\hat{i}+\hat{j}$ are unit vectors.
$\Rightarrow x^2+y^2+z^2=1$ -----(2)
and $(x+1{)}^2+(y+1{)}^2+z^2=1$ -----(3)
Subtracting (2) from (3) gives $x+y=-1$
From (1) and (2) $x^2+y^2=1/2$
$\Rightarrow (x-y{)}^2=2(x^2+y^2)-(x+y{)}^2=0$
$\therefore x=\frac{-1}{2};y=\frac{-1}{2}$ and $z=\frac{1}{\sqrt{2}}$
$\therefore \overrightarrow{a}=-\frac{1}{2}i-\frac{1}{2}j+\frac{1}{\sqrt{2}}k$
Hence, option A.