Question

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

Answer 1

Let \[\vec{r}\] be the given vector, and let it make an angles \[\alpha, \beta, \gamma\] with OX, OY, OZ respectively.
Then, its direction cosines are \[\cos \alpha, \cos \beta, \cos \gamma\].
So, direction ratios of \[\vec{r}\] \[= \hat{j} - \hat{k}\] are proportional to \[0, 1, - 1\].
Therefore, direction cosines of \[\vec{r}\] are \[\frac{0}{\sqrt{0 + 1^2 + \left( - 1 \right)^2}} , \frac{1}{\sqrt{0 + 1^2 + \left( - 1 \right)^2}} , \frac{- 1}{\sqrt{0 + 1^2 + \left( - 1 \right)^2}}\]
  or, 
\[0, \frac{1}{\sqrt{2}} , \frac{- 1}{\sqrt{2}}\]
∴ \[\cos \alpha = 0, \cos \beta = \frac{1}{\sqrt{2}} , \cos \gamma = \frac{- 1}{\sqrt{2}}\]

\[\Rightarrow \alpha = \cos^{- 1} \left( 0 \right) , \beta = \cos^{- 1} \left( \frac{1}{\sqrt{2}} \right) , \gamma = \cos^{- 1} \left( \frac{- 1}{\sqrt{2}} \right)\]

\[ \Rightarrow \alpha = \frac{\pi}{2} , \beta = \frac{\pi}{4} , \gamma = \frac{3\pi}{4}\]