Question

Find the angles at which the following vectors are inclined to each of the coordinate axes:
\[4 \hat{i} + 8 \hat{j} + \hat{k}\]

Answer 1

Let \[\vec{r}\] be the given vector, and let it make an angle \[\alpha, \beta, \gamma\]  with OX, OY, OZ  respectively.
Then, its direction cosines are \[\cos \alpha , \cos \beta , \cos \gamma\].
So, direction ratio of \[\vec{r}\] \[= 4 \hat{i} + 8 \hat{j} + \hat{k}\] are proportional to 4, 8, 1 
Therefore, direction ratio of \[\vec{r}\] are \[\frac{4}{\sqrt{4^2 + 8^2 + 1^2}}, \frac{8}{\sqrt{4^2 + 8^2 + 1^2}} , \frac{1}{\sqrt{4^2 + 8^2 + 1^2}}\]  or,
\[\frac{4}{9}, \frac{8}{9}, \frac{1}{9}\]
∴ \[\alpha = \cos^{- 1} \left( \frac{4}{9} \right) , \beta = \cos^{- 1} \left( \frac{8}{9} \right) , \gamma = \cos^{- 1} \left( \frac{1}{9} \right) .\]