Question

Find the coordinates of the foot of perpendicular and the length of the perpendicular drawn from the point P (5, 4, 2) to the line

\(\vec{r}=-\hat{i}+3\hat{j}+\hat{k}+\lambda (2\hat{i}+3\hat{j}-\hat{k}).\) Also find the image of P in this line.

Answer 1

Given line is

It can be written in cartesian from as

Let Q(\(\alpha, \beta, \gamma\)) be the foot of perpendicular drawn from P(5, 4, 2) to the line (i) and P' (\(x_1, y_1,z_1\)) be the image of P on the line(i)

Parallel vector of line (i) \(\vec{b}=2\hat{i}+3\hat{j}-\hat{k}.\)

[Putting value of \(\alpha, \beta, \gamma\) from (ii)]

Hence the coordinates of foot of perpendicular Q are \((2\times 1-1,\,3\times 1+3,\, -1+1),\) i.e., (1, 6, 0)

\(\therefore\) Length of perpendicular

Also, since Q is mid-point of PP'

Therefore required image is (-3, 8, -2).