Find the coordinates of the foot of the perpendicular drawn from the point A(1, 2, 1) to the line joining the points B(1, 4, 6) and C(5, 4, 4).

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

⇒–4:0:2⇒–2:0:1
\begin{array}{l}
\Rightarrow-4: 0: 2 \\
\Rightarrow-2: 0: 1
\end{array}

So, equation of the line in Cartesian form is

x–1–2=y–40=z–61=λ
\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

(–2λ+1–1):(4–2):(λ+6–1)⇒(–2λ):2:(λ+5)
\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

–2(–2λ)+0+λ+5=0⇒4λ+λ=–5⇒λ=–1
\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