Question
If a unit vector $\vec{a}$ makes angles $\frac{\pi}{3}\ \text{with}\ \hat{i},\frac{\pi}{4}\ \text{with}\ \hat{j}$ and an acute angle $\theta$ with $\hat{k},$ then find $\theta$ and hence, the components of $\vec{a}.$
Again, given $\text{Angel between vectors}\ \hat{a}\ \text{and}\ \hat{j}=0\hat{i}+\hat{j}+0\hat{k}\ \text{is}\ \frac{\pi}{4}.$
$\therefore\ \cos\frac{\pi}{4}=\frac{\hat{a}.{\hat{j}}}{|\hat{a}|.|\hat{j}|}\ \Rightarrow\ \frac{1}{\sqrt{2}}=\frac{\text{x}(0)+\text{y}(1)+\text{z}(0)}{(1)(1)}$
$\Rightarrow\ \frac{1}{\sqrt{2}}=\text{y}\ \ ......\text{(iv)}$
Again, given $\text{Angel between vectors}\ \hat{a}\ \text{and}\ \hat{k}=0\hat{i}+0\hat{j}+\hat{k}\ \text{is}\ \theta,\ \text{where}\ \theta\ \text{is acute angle.}$
$\therefore\ \cos\theta=\frac{\hat{a}.{\hat{k}}}{|\hat{a}|.|\hat{k}|}\ $ $\Rightarrow\ \ \cos\theta=\frac{\text{x}(0)+\text{y}(0)+\text{z}(1)}{(1)(1)}$ $\Rightarrow\ \ \cos\theta=\text{z}\ \ \ .......\text{(v)}$ Putting the values of x, y and z in eq. (ii),$\frac{1}{4}+\frac{1}{2}+\cos^2\theta=1$ $\ \Rightarrow\ \ \cos^2\theta=1-\frac{1}{4}-\frac{1}{2}$
$\Rightarrow\ \ \ \ \ \ \ \cos^2\theta=\frac{4-1-2}{4}=\frac{1}{4}$ $\ \Rightarrow\ \ \cos\theta=\pm\frac{1}{2}$ Since $\theta$ is acute angle, therefore cos $\theta$ is positive and hence $\frac{1}{2}=\cos\frac{\pi}{3}\ \Rightarrow\ \ \theta=\frac{\pi}{3}$ From eq. (v), $\ \ \text{z}=\cos\theta=\frac{1}{2}$ Putting values of x, y and z in eq. (i), $\ \ \hat{a}=\frac{1}{2}\hat{i}+\frac{1}{\sqrt{2}}\hat{j}+\frac{1}{2}\hat{k}$$\therefore\ \ \text{Components of}\ \hat{a}\ \text{are coefficients of}\ \hat{i},\hat{j},\hat{k}\ \text{in}\ \hat{a}$
$\Rightarrow\ \ \frac{1}{2},\frac{1}{\sqrt{2}},\frac{1}{2}\ \text{and angle}\ \theta=\frac{\pi}{3}$Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.