First slide

Dot product or scalar product of vectors

Question

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

Easy

A

$45^{0}$
B

$60^{0}$
C

$\cos^{- 1} (1 / 3)$
D

$\cos^{- 1} (2 / 7)$

Solution

$Since the vectors a + 2 b and 5 a - 4 b are perpendicular so$
$\begin{array}{l} (a + 2 b) . (5 a - 4 b) = 0 \\ \Rightarrow 5 {|a|}^{2} + 6 a \cdot b - 8 {|b|}^{2} = 0 \\ \Rightarrow 6 a \cdot b = 3 (|a| = |b| = 1) \\ \Rightarrow a \cdot b = \frac{1}{2} \Rightarrow \cos θ = \frac{1}{2} (|a| = |b| = 1) \\ \Rightarrow θ = 60^{0} \end{array}$

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