First slide

Vectors

Question

The value of $λ$ for which two vectors $a = 5 \hat{i} + λ \hat{j} + \hat{k} and b = \hat{i} - 2 \hat{j} + \hat{k}$ are perpendicular to. each other is

Moderate

A

2
B

$-$ 2
C

3
D

$-$ 3

Solution

For two vectors a and b to be perpendicular, a · b = 0
Thus, $(5 \hat{i} + λ \hat{j} + \hat{k}) \cdot (\hat{i} - 2 \hat{j} + \hat{k}) = 0$
$5 (\hat{i} \cdot \hat{i}) - 2 λ (\hat{j} \cdot \hat{j}) + 1 (\hat{k} \cdot \hat{k}) = 0$
$\Rightarrow 5 - 2 λ + 1 = 0$
$\Rightarrow$ $6 - 2 λ = 0 \Rightarrow λ = 3$

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