Find the coordinates of the foot of the perpendicular drawn from the origin to the plane 2 x + 3 y + 4 z

Engineering

Mathematics

Line of Greatest Slope

Question

Find the coordinates of the foot of the perpendicular drawn from the origin to the plane

2 x + 3 y + 4 z - 12 = 0

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Given equation is

2 x + 3 y + 4 z = 12

........(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 (2 λ, 3 λ, 4 λ)

If this point lies on plane(i), then

(2 \times 2 λ) + (3 \times 3 λ) + (4 \times 4 λ) = 12 \Rightarrow 29 N = 12 \Rightarrow λ = \frac{12}{29}

Hence, the required point is

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

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