Question

If a unit vector $$\overrightarrow{a}$$ makes an angle $$\frac{\pi }{3}$$ with $$\hat{i},\frac{\pi }{4}$$ with $$\hat{j}$$  and an acute angle θ with $$\hat{k}$$, then find θ and hence, the components of $$\overrightarrow{a}$$.

Answer 1

The Direction cosines of vector $$\overrightarrow{a}$$ are $$l=\cos \left(\frac{\pi }{3}\right)=\frac{1}{2},m=\cos \left(\frac{\pi }{4}\right)=\frac{1}{\sqrt{2}},n=\cos \theta$$ 
Therefore,  
$$l^2+m^2+n^2=1$$
$$\Rightarrow \frac{1}{4}+\frac{1}{2}+n^2=1$$
$$\Rightarrow n^2=1-\frac{3}{4}$$
$$\Rightarrow n^2=\frac{1}{4}$$
$$\Rightarrow n=\frac{1}{2}[\because \overrightarrow{a}\text{ makes acute angle with }\hat{k}]$$
$$\Rightarrow \cos \theta =\frac{1}{2}$$
$$\Rightarrow \theta =\frac{\pi }{3}$$
Since, $$\overrightarrow{a}$$ is the unit vector.
$$\therefore \overrightarrow{a}=l\hat{i}+m\hat{j}+n\hat{k}$$
$$\Rightarrow \overrightarrow{a}=\frac{1}{2}\hat{i}+\frac{1}{\sqrt{2}}\hat{j}+\frac{1}{2}\hat{k}$$
Hence, components of $$\overrightarrow{a}$$ are $$\frac{1}{2}\hat{i}+\frac{1}{\sqrt{2}}\hat{j}+\frac{1}{2}\hat{k}$$