Distance of the point $P (\vec{p})$ from the line $\vec{r} = \vec{a} + λ \vec{b}$ is

Moderate

A

$|\vec{a} - \vec{p}) + \frac{((\vec{p} - \vec{a}) \cdot \vec{b}) \vec{b}}{| \vec{b} |^{2}}|$
B

$|\vec{b} - \vec{p}) + \frac{((\vec{p} - \vec{a}) \cdot \vec{b}) \vec{b}}{| \vec{b} |^{2}}|$
C

$|\vec{a} - \vec{p}) + \frac{((\vec{p} - \vec{b}) \cdot \vec{b}) \vec{b}}{| \vec{b} |^{2}}|$
D

none of these

Solution

Let $Q (\vec{q})$ be the foot of altitude drawn from $P (\vec{p})$ to the line $\vec{r} = \vec{a} + λ \vec{b}$
$\begin{array}{l} \Rightarrow (\vec{q} - \vec{p}) \cdot \vec{b} = 0 and \vec{q} = \vec{a} + λ \vec{b} \\ \Rightarrow (\vec{a} + λ \vec{b} - \vec{p}) \cdot \vec{b} = 0 \\ or (\vec{a} - \vec{p}) \cdot \vec{b} + λ | \vec{b} |^{2} = 0 \\ or λ = \frac{(\vec{p} - \vec{a}) \cdot \vec{b}}{| \vec{b} |^{2}} \\ \Rightarrow \vec{q} - \vec{p} = \vec{a} + \frac{((\vec{p} - \vec{a}) \cdot \vec{b}) \vec{b}}{| \vec{b} |^{2}} - \vec{p} \\ \Rightarrow | \vec{q} - \vec{p} | = |\vec{a} - \vec{p}) + \frac{((\vec{p} - \vec{a}) \cdot \vec{b}) \vec{b}}{| \vec{b} |^{2}}| \end{array}$

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