Question

Show that the vector $\overrightarrow{a}=(\hat{i}+\hat{j}+\hat{k})$ is equally inclined to the coordinate axes.

Answer 1

Lets find the angle formed by the vector and the axes

$x\to iy\to jz\to \hat{k}$

$\to {\theta }_1\to {\theta }_2\to {\theta }_3$

$\cos {\theta }_1=\frac{\hat{i}\left(\hat{i}+\hat{j}+\hat{k}\right)}{\sqrt{3}}=\frac{1}{\sqrt{3}}$

$\theta \text{=}\text{co}{\text{s}}^{-1}\left(\frac{1}{\sqrt{3}}\right)$

$\cos {\theta }_2=\frac{\text{j}\left(\hat{i}+\hat{j}+k\right)}{\sqrt{3}}$

${\theta }_2\text{=}\text{co}{\text{s}}^{-1}\left(\frac{1}{\sqrt{3}}\right)$

$\cos {\theta }_3=\frac{\hat{k}\left(\hat{i}+\hat{j}+\hat{k}\right)}{\sqrt{3}}$

${\theta }_3=\text{co}{\text{s}}^{-1}\left(\frac{1}{\sqrt{3}}\right)$

${\theta }_1={\theta }_2={\theta }_3$

$so\text{vector in equally inclined with the coordinate axes}$