Question

If the vector $\mathbf{i} + \mathbf{j} + \mathbf{k}$ makes angles $\alpha,\beta,\gamma$with vectors $\mathbf{i},\mathbf{j},\mathbf{k}$respectively, then

• A. $\alpha = \beta \neq \gamma$
• B. $\alpha = \gamma \neq \beta$
• C. $\beta = \gamma \neq \alpha$
• D. $\alpha = \beta = \gamma$

Answer 1

Answer: (D)

$\alpha = \beta = \gamma$

Answer 2

Angle between $\mathbf{i} + \mathbf{j} + \mathbf{k}$ and $\mathbf{i}$ is equal to

$\cos^{- 1}\left\{ \frac{(\mathbf{i} + \mathbf{j} + \mathbf{k}).\mathbf{i}}{|\mathbf{i} + \mathbf{j} + \mathbf{k}||\mathbf{i}|} \right\} \Rightarrow \alpha = \cos^{- 1}\left( \frac{1}{\sqrt{3}} \right)$

Similarly angle between $\mathbf{i} + \mathbf{j} + \mathbf{k}$ and $\mathbf{j}$ is $\beta = \cos^{- 1}\left( \frac{1}{\sqrt{3}} \right)$

and between $\mathbf{i} + \mathbf{j} + \mathbf{k}$ and $\mathbf{k}$is $\gamma = \cos^{- 1}\left( \frac{1}{\sqrt{3}} \right).$

Hence $\alpha = \beta = \gamma.$