If a and b_1 are two unit vectors such that a + 2b and 5a - 4b, are perpendicular to each other, then the angle between a and b is

If a and b_1 are two unit vectors such that a + 2b and 5a -… — JEE Main 2002 Mathematics PYQ

AIEEE 2002

\vec{a}

and

\vec{b_1}

are two unit vectors such that

\vec{a} + 2\vec{b}

and

5\vec{a} - 4\vec{b}

, are perpendicular to each other, then the angle between

\vec{a}

and

\vec{b}

A.
45°
B.
60°
C.
$\cos^{-1}\left(\dfrac{1}{3}\right)$
D.
$\cos^{-1}\left(\dfrac{2}{7}\right)$

Show answer and solution

Correct option: B

Solution

Since (a + 2b) ⊥ (5a - 4b), their dot product is zero:

(a + 2b)·(5a - 4b) = 5|a|² - 4a·b + 10a·b - 8|b|² = 0

Since both are unit vectors, |a|² = |b|² = 1:

5 - 4a·b + 10a·b - 8 = 0 → 6a·b - 3 = 0 → a·b = 1/2

Therefore cosθ = 1/2, giving θ = 60°.

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