Question

A unit vector making angle $\frac{\mathrm{\pi }}{4}$ with x-axis, $\frac{\mathrm{\pi }}{3}$ with y-axis and an acute angle with z-axis is ______________.

Answer 1

Given, a unit vector makes angle $\frac{\mathrm{\pi }}{4}$ with x-axis, $\frac{\mathrm{\pi }}{3}$ with y-axis
i.e l = cos$\frac{\mathrm{\pi }}{4}$, m = cos$\frac{\mathrm{\pi }}{3}$, n = cos$\gamma$ (say)
Since l² + m² + n² = 1

$$\begin{gathered} \mathrm{i}.\mathrm{e}{\left(\cos \frac{\mathrm{\pi }}{4}\right)}^2+{\left(\cos \frac{\mathrm{\pi }}{3}\right)}^2+{\cos }^2\gamma =1 \\ \mathrm{i}.\mathrm{e}{\left(\frac{1}{\sqrt{2}}\right)}^2+{\left(\frac{1}{2}\right)}^2+{\cos }^2\gamma =1 \end{gathered}$$
$\begin{array}{rcl}\mathrm{i}.\mathrm{e}{\cos }^2\gamma & =& 1-\frac{1}{2}-\frac{1}{4}\\ & =& \frac{4-2-1}{4}=\frac{1}{4}\\ \mathrm{i}.\mathrm{e}\cos \gamma & =& \pm \frac{1}{2}\end{array}$
Since γ is given to be acute
$$\begin{gathered} \therefore \cos \gamma =\frac{1}{2} \\ \therefore \mathrm{vector}\mathrm{is}\frac{1}{\sqrt{2}}\hat{i}+\frac{1}{2}\hat{j}+\frac{1}{2}\hat{k} \end{gathered}$$