Question

Write down a unit vector in XY- plane, making an angle of ${30}^{\circ }$ in anticlockwise direction with the positive direction of X-axis.

Answer 1

Let OP make ${30}^{\circ }$ with X-aixs, ${60}^{\circ }$ with Y-axis and ${90}^{\circ }$ with Z-axis and it lies in XY-plane.
$\therefore$ Direction cosines of OP are cos ${30}^{\circ }$, cos ${60}^{\circ }$ and co ${90}^{\circ }$ i.e.,$\frac{\sqrt{3}}{2},\frac{1}{2},0.\text{Thus,OP}is\frac{\sqrt{3}}{2}\hat{i}+\frac{1}{2}\hat{j},$
$|OP|=\sqrt{\left({\frac{\sqrt{3}}{2}}^2\right)+{\left(\frac{1}{2}\right)}^2}=\sqrt{\frac{3}{4}+\frac{1}{4}}=\sqrt{1}=1$
which is required unit vector in XY-plane.