Find the distance of the point P(6,5,9) from the plane determined by the points A(3,−1,2),B(5,2,4) and C(−1,−1,−6)

Engineering

Mathematics

Plane and Its Different Forms

Equation of Straight Line in 3D

Distance from a Plane

Question

Find the distance of the point P(6,5,9) from the plane determined by the points A(3,−1,2),B(5,2,4) and C(−1,−1,−6)

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Equation of plane is 3 points from

$|\begin{array}{ccc} x - 3 & y + 1 & z - 2 \\ 2 & 3 & 2 \\ - 4 & 0 & 4 \end{array}| = 0$

⇒ 3x – 4y + 3z – 19 = 0

∴ distance from (6,5,9) is

$\frac{3 \times 6 - 4 \times 5 + 3 \times 9 - 19}{\sqrt{9 + 16 + 9}}$

$= \frac{18 - 20 + 27 + 19}{\sqrt{34}} = \frac{6}{\sqrt{34}} units.$