If two vectors and are perpendicular to each other. Then, the value of m will be [2023]

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Solution

Q.
If two vectors $\vec{P} = \hat{i} + 2 m \hat{j} + m \hat{k}$ and $\vec{Q} = 4 \hat{i} - 2 \hat{j} + m \hat{k}$ are perpendicular to each other. Then, the value of m will be [2023]

1 –1

2 2

3 3

4 1

Ans.
(2)

$\vec{P} \cdot \vec{Q} = 0$

$(\hat{i} + 2 m \hat{j} + m \hat{k}) \cdot (4 \hat{i} - 2 \hat{j} + m \hat{k}) = 0 \Rightarrow 4 - 4 m + m^{2} = 0$

$\Rightarrow {(m - 2)}^{2} = 0 \Rightarrow m = 2$

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