Question

A unit vector in $xy$ - plane makes an angle of $45^{\circ }$ with the vector $\hat{i}+\hat{j}$ and an angle of $60^{\circ }$ with the vector $3\hat{i}-4\hat{j}$ is

• A. $\frac{13}{7}\hat{i}+\frac{1}{7}\hat{j}$
• B. $\frac{7}{13}\hat{i}+\frac{7}{15}\hat{j}$
• C. $\frac{13}{14}\hat{i}++\frac{1}{14}\hat{j}$
• D. None of the above

Answer 1

Answer: (C)

Let the required unit vector be $r=a\hat{i}+b\hat{j}$.
Then, $|r|=1$
$\Rightarrow a^2+b^2=\mathrm{1...}$(i)
Since, $r$ makes an angle of $45^{\circ }$ with $\hat{i}+\hat{j}$ and an angle of $60^{\circ }$ with $3\hat{i}-4\hat{j}$, therefore,
$\cos \frac{\pi }{4}=\frac{r\cdot (\hat{i}+\hat{j})}{|r||\hat{i}+\hat{j}|}$
and $\cos \frac{\pi }{3}=\frac{r\cdot (3\hat{i}-4\hat{j})}{|r||3\hat{i}-4\hat{j}|}$
$\Rightarrow \frac{1}{\sqrt{2}}=\frac{a+b}{\sqrt{2}}$
and $\frac{1}{2}=\frac{3a-4b}{5}$
$\Rightarrow a+b=1$
and $a-4b=\frac{5}{2}$
$\Rightarrow a=\frac{13}{14},b=\frac{1}{14}$
$\therefore r=\frac{13}{14}\hat{i}+\frac{1}{14}\hat{j}$