Question

if Q be the foot of perpendicular from p(2,4,3) on the line joining the points a(1,2,4) and b(3,4,5) .find the coordinate of q.

Answer 1

3 dim lines coordinate geometry.

Points are P(2,4,3). A(1,2,4). B(3,4,5). To find the foot of the perpendicular from P onto AB.

Equation of AB:
x = 1+(3-1)t = 1+2t.
y = 2+(4-2)t = 2+2t
z = 4+(5-4)t = 4+t
Direction Coefficient s: 2 2 1. Q lies on the AB.

Equation of a 3-d plane perpendicular to AB:
2x + 2y + z = C

This plane passes through P.
So 2*2+2*4+3= C.

Q lies on the line and the plane.
So 4t+2 + 4t+4 + t+4 = 15.
So for Q: t = 5/9.
Ie. Q=(19/9, 28/9, 41/9).