Question

Find a vector \[\overrightarrow{a}\] of magnitude \[5\sqrt{2}\], making an angle of \[\frac{\pi}{4}\] with x-axis, \[\frac{\pi}{2}\] with y-axis and an acute angle θ with z-axis. 

Answer 1

It is given that vector \[\overrightarrow{a}\] makes an angle of \[\frac{\pi}{4}\] with x-axis, \[\frac{\pi}{2}\] with y-axis and an acute angle θ with z-axis.
\[\therefore l = \cos\frac{\pi}{4} = \frac{1}{\sqrt{2}}, m = \cos\frac{\pi}{2} = 0, n = \cos\theta\]
Now,

\[l^2 + m^2 + n^2 = 1\]

\[ \Rightarrow \frac{1}{2} + 0 + \cos^2 \theta = 1\]

\[ \Rightarrow \cos^2 \theta = 1 - \frac{1}{2} = \frac{1}{2}\]

\[ \Rightarrow \cos\theta = \frac{1}{\sqrt{2}} \left( \theta\text{ is acute }\right)\]
We know that 
\[\overrightarrow{a} = \left| \overrightarrow{a} \right|\left( l \hat{i} + m \hat{j} + n \hat{k} \right)\]
\[ \Rightarrow \overrightarrow{a} = 5\sqrt{2}\left( \frac{1}{\sqrt{2}} \hat{i} + 0 \hat{j} + \frac{1}{\sqrt{2}} \hat{k} \right)\]
\[ \Rightarrow \overrightarrow{a} = 5\left( \hat{i} + 0 \hat{j} + \hat{k} \right)\]