Find the coordinates of the foot of the perpendicular drawn from the origin to the plane 2x + 3y + 4z − 12 = 0.

Engineering

Mathematics

Plane and Its Different Forms

Distance from a Plane

Question

Find the coordinates of the foot of the perpendicular drawn from the origin to the plane 2x + 3y + 4z − 12 = 0.

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Given equation is 2x + 3y + 4z − 12 = 0 ........(i)

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

Equation of the line through O(0,0,0) and perpendicular to plane (i) is

\frac{x}{2} = \frac{y}{3} = \frac{z}{4} = λ

(say) ........(ii)

A general point on (ii) is N(2λ,3λ,4λ)

If this point lies on plane(i), then

(2 × 2λ) + (3 × 3λ) + (4 × 4λ) = 12

\Rightarrow 29 N = 12 \Rightarrow λ = \frac{12}{29}

Hence, the required point is

(\frac{24}{29}, \frac{36}{29}, \frac{48}{29})