NDA 1 2025 Maths, question 70

Topic: Vectors

Q70

Let $\overrightarrow{a}$ , $\overrightarrow{b}$ , $\left( \overrightarrow{a} \times \overrightarrow{b} \right)$ be unit vectors. What is $\left( \overrightarrow{a} \cdot \overrightarrow{b} \right)$ equal to?

(a) $0$ Correct answer
(b) $1/2$
(c) $1$
(d) $3$

Solution.

Answer: (a) $0$

Use $\left| \overrightarrow{a} \times \overrightarrow{b} \right|^{2} + \left( \overrightarrow{a} \cdot \overrightarrow{b} \right)^{2} = \left| \overrightarrow{a} \right|^{2}\left| \overrightarrow{b} \right|^{2}$ .
Each given magnitude is 1, so $1 + \left( \overrightarrow{a} \cdot \overrightarrow{b} \right)^{2} = 1$ .
Thus $\overrightarrow{a} \cdot \overrightarrow{b} = 0$ : the two vectors are perpendicular.