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

Two vectors $$\vec{P}$$ and $$\vec{Q}$$ are perpendicular if their dot product is zero:

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

Given:

$$\vec{P} = \hat{i} + 2m\hat{j} + m\hat{k}$$

$$\vec{Q} = 4\hat{i} - 2\hat{j} + m\hat{k}$$

Computing the dot product:

$$\vec{P} \cdot \vec{Q} = (1)(4) + (2m)(-2) + (m)(m) = 0$$

$$4 - 4m + m^2 = 0$$

$$m^2 - 4m + 4 = 0$$

$$(m - 2)^2 = 0$$

$$m = 2$$

The correct answer is Option 2: $$m = 2$$.