Find the coordinates of the foot of the perpendicular drawn from the point A(1,8,4) to the line joining the points B(0,−1,3) and C(2,−3,−1).

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(1,8,4) to the line joining the points B(0,−1,3) and C(2,−3,−1).

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Co-ordinates of B = (0,–1,3) & C = (2,–3,–1)

D.r's of BC = 2,–2,–4

cartesion equation of BC

$= \frac{x - 0}{2} = \frac{y + 1}{- 2} = \frac{z - 3}{- 4} = λ$

Any point on bC will have co-ordinate

= (2λ,–2λ – 1,–4λ + 3)

so, D = (2λ,–2λ – 1,–4λ + 3)
Dir's of AD

= (2λ – 1, –2λ – 9,–4λ – 1)

since, AD ⊥r BC

⇒ (2λ – 1)(2) + (–2λ – 9)(–2) + (–4λ –1)(–4) = 0

$∴ λ = - \frac{5}{6}$

So, co-ordinate of foot of ⊥r

$= (- \frac{5}{3}, \frac{2}{3}, \frac{19}{3})$ .