the vector of length l is turned through the angle theta about its tail. what is the change in the position vector of its head?
options are
1)l cos(@/2)
2) 2lsin(@/2)
3) 2lcos(@/2)
4) lsin(@/2)
where @ stands for theta...... experts please help......

In the above figure A is the original vector. After rotating by the tail by angle $\theta$ it becomes A'. Now since vector is defined by only length and direction, not by it's initial position, so the A'₁ are the same vector. Now the position vector of the head of the vector is $\overrightarrow{A}{\text{'}}_1=l\cos \theta \hat{i}+l\sin \theta \hat{j}$, and the magnitude will be l, since because of the rotation magnitude will not change.
Now your given option neither represent any vector nor it will be the magnitude of the vector, so answer will not match with any of the given option.