Find the coordinates of the foot of the perpendicular drawn from the point to the line joining the points and .

Answer

Given: perpendicular drawn from point to line joining points and

To find: foot of perpendicular

Formula Used: Equation of a line is

Vector form:

Cartesian form:

where is a point on the line and with being the direction ratios of the line.

If 2 lines of direction ratios and are perpendicular, then

Explanation:

is a point on the line.

Therefore,

Also direction ratios of the line are

\begin{array}{l}

\Rightarrow-4: 0: 2 \\

\Rightarrow-2: 0: 1

\end{array}

So, equation of the line in Cartesian form is

$\frac{\mathrm{x}\u20131}{\u20132}=\frac{\mathrm{y}\u20134}{0}=\frac{\mathrm{z}\u20136}{1}=\lambda $\frac{\mathrm{x}-1}{-2}=\frac{\mathrm{y}-4}{0}=\frac{\mathrm{z}-6}{1}=\lambda

Any point on the line will be of the form

So the foot of the perpendicular is of the form

The direction ratios of the perpendicular is

\begin{array}{l}

(-2 \lambda+1-1):(4-2):(\lambda+6-1) \\

\Rightarrow(-2 \lambda): 2:(\lambda+5)

\end{array}

From the direction ratio of the line and the direction ratio of its perpendicular, we have

$\begin{array}{l}\u20132(\u20132\lambda )+0+\lambda +5=0\\ \Rightarrow 4\lambda +\lambda =\u20135\\ \Rightarrow \lambda =\u20131\end{array}$\begin{array}{l}

-2(-2 \lambda)+0+\lambda+5=0 \\

\Rightarrow 4 \lambda+\lambda=-5 \\

\Rightarrow \lambda=-1

\end{array}

So, the foot of the perpendicular is