Question

Find the foot of the perpendicular from (1, 2, −3) to the line \[\frac{x + 1}{2} = \frac{y - 3}{- 2} = \frac{z}{- 1} .\]

Answer 1

Let L be the foot of the perpendicular drawn from the point P (1, 2, -3) to the given line. The coordinates of a general point on the line \[\frac{x + 1}{2} = \frac{y - 3}{- 2} = \frac{z}{- 1}\]  are given by 

\[\frac{x + 1}{2} = \frac{y - 3}{- 2} = \frac{z}{- 1} = \lambda\]

\[ \Rightarrow x = 2\lambda - 1\]

\[ y = - 2\lambda + 3 \]

\[ z = - \lambda\] 

Let the coordinates of L be  \[\left( 2\lambda - 1, - 2\lambda + 3 , - \lambda \right)\] 

 

The direction ratios of PL are proportional to \[2\lambda - 1 - 1, - 2\lambda + 3 - 2, - \lambda + 3, i . e . 2\lambda - 2, - 2\lambda + 1, - \lambda + 3\]  The direction ratios of the given line are proportional to 2, -2,-1,  but PL is perpendicular to the given line. 

\[\therefore 2\left( 2\lambda - 2 \right) - 2\left( - 2\lambda + 1 \right) - 1\left( - \lambda + 3 \right) = 0\]

\[ \Rightarrow \lambda = 1\] 
Substituting  

\[\lambda = 1\] in 

\[\left( 2\lambda - 1, - 2\lambda + 3 , - \lambda \right)\] 

we get the coordinates of L as (1,1,-1).