Question

In the following case, find the coordinates of the foot of the perpendicular drawn from the origin:
x +y +z =1

Answer 1

Given plane is x+y+z=1 ...(iii)

$\therefore$ Dr's of any line perpendicular to plane (iii) are (1,1,1)

Hence, equation of the line through origin and at right angle to Eq. (iii) are $\frac{x-0}{1}=\frac{y-0}{1}=\frac{z-0}{1}$ ...(iv)

Any point on this line is (t,t,t). This point lies in the plane , if

t+t+t=1 ie., 3t=1 $\Rightarrow t=\frac{1}{3}$

Hence, the reqired foot of the perpendicular is $(\frac{1}{3},\frac{1}{3},\frac{1}{3}).$