Question

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

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 ratios of \[\vec{r}\] \[\hat{i} - \hat{j} + \hat{k}\] are proportional to 1, - 1, 1.
Therefore, Direction cosine of \[\vec{r}\] are  \[\frac{1}{\sqrt{1^2 + \left( - 1 \right)^2 + 1^2}} , \frac{- 1}{\sqrt{1^2 + \left( - 1 \right)^2 + 1^2}} , \frac{1}{\sqrt{1^2 + \left( - 1 \right)^2 + 1^2}}\] or,
\[\frac{1}{\sqrt{3}}, \frac{- 1}{\sqrt{3}}, \frac{1}{\sqrt{3}}\]
∴ \[\cos \alpha = \frac{1}{\sqrt{3}}, \cos \beta = \frac{- 1}{\sqrt{3}}, \cos \gamma = \frac{1}{\sqrt{3}}\]
\[\alpha = \cos^{- 1} \left( \frac{1}{\sqrt{3}} \right) , \beta = \cos^{- 1} \left( \frac{- 1}{\sqrt{3}} \right) , \gamma = \cos^{- 1} \left( \frac{1}{\sqrt{3}} \right)\]