Question

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

• A. $\overline{i}$
• B. $(\overline{i}+\overline{j})/\sqrt{2}$
• C. $(\overline{i}-\overline{j})/\sqrt{2}$
• D. no such vector is exists

Answer 1

Answer: (D)

no such vector is exists

Let vector be $ai+bj$

Now,

$\overrightarrow{a}.\overrightarrow{b}=|\overrightarrow{a}||\overrightarrow{b}|\cos \theta$

$a.1+b.1=1\left(\sqrt{2}\right)\cos 45$

$a+b=1$

And, $a^2+b^2=1$

$\Rightarrow (a+b{)}^2-a^2{b}^2=0$

$\Rightarrow 2ab=0$

So, $a=0$ or $b=0$

$a=1$ or $b=0$

$a=0$ or $b=1$

$a.3\pm b.4=5\cos 60$

$\Rightarrow 3a-4b=\frac{5}{2}$

Thus, neither $a=1$ nor $b=1$ suit.

So, there is no vector in xy plane which satisfies given condition.