NDA 1 2025 Maths, question 66

Topic: Analytical geometry

Q66

A line makes angles $\alpha$ , $\beta$ and $\gamma$ with the positive directions of the coordinate axes. If $\overrightarrow{a} = \left( \sin^{2}\alpha \right)\widehat{i} + \left( \sin^{2}\beta \right)\widehat{j} + \left( \sin^{2}\gamma \right)\widehat{k}$ and $\overrightarrow{b} = \widehat{i} + \widehat{j} + \widehat{k}$ , then what is $\overrightarrow{a} \cdot \overrightarrow{b}$ equal to?

(a) $- 2$
(b) $- 1$
(c) $1$
(d) $2$ Correct answer

Solution.

Answer: (d) $2$

The dot product is $\sin^{2}\alpha + \sin^{2}\beta + \sin^{2}\gamma$ .
Direction cosines obey $\cos^{2}\alpha + \cos^{2}\beta + \cos^{2}\gamma = 1$ .
Therefore the required sum is $3 - 1 = 2$ .