Find the equation of a straight line in the plane $\vec{r} \cdot \vec{n} = d$ which parallel to $\vec{r} = \vec{a} + λ \vec{b}$ and passes through the foot of the perpendicular drawn from point $P (\vec{a}) to \vec{r} \cdot \vec{n} = d (where \vec{n} \cdot \vec{b} = 0)$

Moderate

A

$\vec{r} = \vec{a} + (\frac{d - \vec{a} \cdot \vec{n}}{n^{2}}) \vec{n} + λ \vec{b}$
B

$\vec{r} = \vec{a} + (\frac{d - \vec{a} \cdot \vec{n}}{n}) \vec{n} + λ \vec{b}$
C

$\vec{r} = \vec{a} + (\frac{\vec{a} \cdot \vec{n} - d}{n^{2}}) \vec{n} + λ \vec{b}$
D

$\vec{r} = \vec{a} + (\frac{\vec{a} \cdot \vec{n} - d}{n}) \vec{n} + λ \vec{b}$

Solution

Foot of the perpendicular from point $A (\vec{a})$ on the plane $\vec{r} \cdot \vec{n} = d is \vec{a} + \frac{(d - \vec{a} \cdot \vec{n})}{| \vec{n} |^{2}} \vec{n}$
Therefore, equation of the line parallel to $\vec{r} = \vec{a} + λ \vec{b}$ in the plane $\vec{r} \cdot \vec{n} = d$ is given by
$\vec{r} = \vec{a} + \frac{(d - \vec{a} \cdot \vec{n})}{| \vec{n} |^{2}} \vec{n} + λ \vec{b}$

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