Question
From a point P(1, 2, 4), a perpendicular is drawn on the plane 2x + y – 2z + 3 = 0. Find the equation, the length and coordinates of the foot of the perpendicular.
Solution
The equation of plane is
2x + y – 2z + 3 = 0 ...(1)
Direction ratios of the normal to the plane are 2, 1, – 2.
Let M be the foot of perpendicular from P(1, 2, 4) to the plane.
Now PM is a straight line which passes through P(1, 2, 4) and has direction ratios 2, 1,–2.
∴ its equations are
Any point M on line is. (2 r + 1, r + 2, – 2 r + 4)
∴ M lies on plane (1)
∴ 2 (2 r + 1) + (r + 2) – 2 (–2 r + 4) + 3 = 0
∴ 4 r + 2 + r + 2 + 4 r – 8 + 3 = 0
Length of perpendicular 
