Find the coordinates of the foot of the perpendicular drawn from the point
Answer
Given: perpendicular drawn from point
To find: foot of perpendicular
Formula Used: Equation of a line is
Vector form:
Cartesian form:
where
If 2 lines of direction ratios
Explanation:
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
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
\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