GATE 2022 AE – Question 12

Correct answer: (D) $-\sqrt2\hat i$

Explanation

Rotating a vector in the $xy$-plane about the positive $z$-axis by an angle $\theta$ (counter-clockwise) uses the rotation matrix
$$\begin{pmatrix}x^\prime\\y^\prime\end{pmatrix}=\begin{pmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix}.$$

For $\hat i+\hat j=(1,1)$ and $\theta=135^\circ$ ($\cos135^\circ=-\tfrac{\sqrt2}{2}$, $\sin135^\circ=\tfrac{\sqrt2}{2}$):
$$x^\prime=\cos\theta-\sin\theta=-\tfrac{\sqrt2}2-\tfrac{\sqrt2}2=-\sqrt2,$$
$$y^\prime=\sin\theta+\cos\theta=\tfrac{\sqrt2}2-\tfrac{\sqrt2}2=0 .$$

The result is $-\sqrt2\,\hat i$ (option D).

(Check: the vector $\hat i+\hat j$ has length $\sqrt2$ and points at $45^\circ$. Adding $135^\circ$ gives $180^\circ$, i.e. along $-\hat i$ with the same length $\sqrt2$.)