Question

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

• A. $i$
• B. $(a+j)/\sqrt{2}$
• C. $(i-j)/\sqrt{2}$
• D. None of these

Answer 1

The correct option is D None of these

Let the required unit vector on the $x-y$ plane be $r=xi+yj$

$\therefore \left|r\right|=\sqrt{x^2+y^2}=1\Rightarrow x^2+y^2=1$ ...(1)

Now angle between $r$ and the vector $i+j$ is ${45}^{\circ }$
and the angle between $r$ and the vector $3i+4j$ is ${60}^{\circ }$

$\therefore \cos {45}^{\circ }=\frac{(xi+yj).(i+j)}{|xi+yj||i+j|}$

And $\cos {60}^{\circ }=\frac{(xi+yj)\cdot (3i-4j)}{|xi+yj||3i-4j|}$

$\Rightarrow x+y=1$ ...(2)

$3x-4y=\frac{5}{2}$ ...(3)

There exists no real value of $x$ and $y$ satisfying equation (1) , (2) and (3)
Hence correct answer is (d)