NIMCET 2014 — Mathematics PYQ

Solution:

Let $\vec{a}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k}$

Given $|\vec{a}|=3$

$\sqrt{a_1^2+a_2^2+a_3^2}=3$

$a_1^2+a_2^2+a_3^2=9$ ....(1)

Let the equal angle be $\alpha$.

The direction cosines are $l=\cos \alpha ,m=\cos \alpha ,n=\cos \alpha$.

We know that:

$l^2+m^2+n^2=1$

${\cos }^2\alpha +{\cos }^2\alpha +{\cos }^2\alpha =1$

$3{\cos }^2\alpha =1$

${\cos }^2\alpha =\frac{1}{3}$

$\cos \alpha =\frac{1}{\sqrt{3}}$

$\alpha ={\cos }^{-1}\left(\frac{1}{\sqrt{3}}\right)$

Correct Option: 1