Question

Find a vector $\overrightarrow{a}$ of magnitude $5\sqrt{2}$, making an angle of $\frac{\mathrm{\pi }}{4}$ with x-axis, $\frac{\mathrm{\pi }}{2}$ with y-axis and an acute angle θ with z-axis. [CBSE 2014]

Answer 1

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

$$\begin{gathered} l^2+m^2+n^2=1 \\ \Rightarrow \frac{1}{2}+0+{\cos }^2\theta =1 \\ \Rightarrow {\cos }^2\theta =1-\frac{1}{2}=\frac{1}{2} \\ \Rightarrow \cos \theta =\frac{1}{\sqrt{2}}\left(\theta \mathrm{is}\mathrm{acute}\right) \end{gathered}$$

We know that

$$\begin{gathered} \overrightarrow{a}=\left|\overrightarrow{a}\right|\left(l\hat{i}+m\hat{j}+n\hat{k}\right) \\ \Rightarrow \overrightarrow{a}=5\sqrt{2}\left(\frac{1}{\sqrt{2}}\hat{i}+0\hat{j}+\frac{1}{\sqrt{2}}\hat{k}\right) \\ \Rightarrow \overrightarrow{a}=5\left(\hat{i}+0\hat{j}+\hat{k}\right) \end{gathered}$$