Find the distance between the point (7, 2, 4) and the plane determined by the points A(2, 5, –3), B(–2, –3, 5) and (5, 3, –3).
The equation of the plane passing through (x1, y1, z1), (x2, y2, z2) and (x3, y3, z3) is given by the following equation:
According to question,
(x1, y1, z1) = (2, 5, –3)
(x2, y2, z2) = (–2, –3, 5)
(x3, y3, z3) = (5, 3, –3)
Putting these values,
⟹
⟹ (x – 2)(16) + (y – 5)(24) + (z + 3)(32) = 0
⟹ 2x + 3y + 4z –7 = 0
Distance of 2x + 3y + 4z –7 = 0 from (7, 2, 4)
We know, the distance of point (x1,y1,z1) from the plane
is given by:
⟹
⟹ p = √29 units
∴ the distance between the point (7, 2, 4) and the plane determined by the points A(2, 5, –3), B(–2, –3, 5) and (5, 3, –3) is √29 units.