NDA 1 2024 Maths, question 70

Topic: Vectors

Q70

Let $\overrightarrow{a}$ and $\overrightarrow{b}$ are two vectors of magnitude $4$ inclined at an angle $\frac{\pi}{3}$ , then what is the angle between $\overrightarrow{a}$ and $\overrightarrow{a}-\overrightarrow{b}$ ?

(a) $\frac{\pi}{2}$
(b) $\frac{\pi}{3}$ Correct answer
(c) $\frac{\pi}{4}$
(d) $\frac{\pi}{6}$

Solution.

Answer: (b) $\frac{\pi}{3}$

The data give |a − b| = 4 and a·(a − b) = |a|² − a·b = 16 − 8 = 8.
Therefore the cosine of the angle between a and a − b is 8/(4 × 4) = 1/2, so the angle is π/3.