Find the coordinates of the foot of the perpendicular drawn from the point A(1,2,1) to the line joining the points B(1,4,6) and C(5,4,4).

Engineering

Mathematics

Equation of Straight Line in 3D

Distance from a Plane

Question

Find the coordinates of the foot of the perpendicular drawn from the point A(1,2,1) to the line joining the points B(1,4,6) and C(5,4,4).

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D.r's pf line BC

= (5 –1), (4 – 4), (4 – 6)

= 4, 0,–2

cartesion equation of line BC

$= \frac{X - 1}{4} = \frac{Y - 4}{0} = \frac{Z - 6}{- 2} = λ$

Since D lies on BC

D = (4λ + 1, 4, –2λ + 6) ------- (1)

D.r's of AD are

Now, AD ⊥ BC

So, $\vec{AD}, \vec{BC} = 0$

∴ 4 (4λ) + 0(2) + (–2) (–2λ + 5) = 0

⇒ λ = 1/2

So, co-ordinates of D are

(3,4,5).