Physics - Vectors Question with Solution | TestHub

For an observer on the $+y$ axis looking towards the origin, the normal to the plane of observation is $\hat{j}$.

By the right-hand rule, a counter-clockwise rotation in the $x-z$ plane corresponds to curling fingers from the $+z$ axis to the $+x$ axis (since $\hat{k}\times \hat{i}=\hat{j}$).

Therefore, a clockwise rotation is in the opposite direction, from the $+x$ axis towards the $+z$ axis.

The initial vector is $\vec{v}=3\hat{i}$, which lies along the $+x$ axis.

When it is rotated clockwise by an angle $\theta$, it moves towards the $+z$ axis. The new vector will have a positive $x$-component and a positive $z$-component.

From the geometry, the component along the $x$-axis becomes $3\cos \theta$ and the component along the $z$-axis becomes $3\sin \theta$.

Thus, the rotated vector is ${\vec{v}}^{\prime }=3\cos \theta \hat{i}+3\sin \theta \hat{k}$.

Answer: $3\cos \theta \hat{i}+3\sin \theta \hat{k}$