Determine the co-ordinates of the foot of the perpendicular drawn from the origin to the plane 4x - 2y + 3z - 6 = 0

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AAI ATC Junior Executive 25 March 2021 Official Paper (Shift 2)
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Given:

The plane : 4x - 2y + 3z - 6 = 0

Concept:

For a plane having equation ax + by + cz = d(a, b, c) are direction ratios of the normal to the plane.

Equation of a line passing through (x1, y1, z1) and having direction ratios a, b, c in the cartesian form :

Calculation:

Direction ratios of normal to the plane are 4, -2, and 3. 

Equation of a line passing through the origin (0, 0, 0) and having direction ratios 4, -2, 3 is:

⇒ x = 4λ , y = -2λ and z = 3λ

Satisfying equation of the plane with above coordinates :

⇒ 4(4λ) - 2(-2λ) + 3(3λ) - 6 = 0

⇒ λ = 6/29

∴ Foot of the perpendicular,

⇒ x = 24/29, y = -12/29 and z = 18/29

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