Consider a plane passing through the points A(2, 2, 1) , B(3, 4, 2) and C(7, 0, 6) . Which one of the followin

Question

Consider a plane passing through the points A(2, 2, 1) , B(3, 4, 2) and C(7, 0, 6) . Which one of the following points lies on this plane?

Accepted Answer

The equation of a plane passing through

(x_{1}, y_{1}, z_{1}), (x_{2}, y_{2}, z_{2}), (x_{3}, y_{3}, z_{3})

is:

x - x_{1} x_{2} - x_{1} x_{3} - x_{1} y - y_{1} y_{2} - y_{1} y_{3} - y_{1} z - z_{1} z_{2} - z_{1} z_{3} - z_{1} = 0

x - 2 3 - 2 7 - 2 y - 2 4 - 2 0 - 2 z - 1 2 - 1 6 - 1 = 0

x - 2 15 y - 2 2 - 2 z - 1 15 = 0

Expanding along

R_{1}

:

(x - 2) (10 - (- 2)) - (y - 2) (5 - 5) + (z - 1) (- 2 - 10) = 0

12 (x - 2) - 0 (y - 2) - 12 (z - 1) = 0

12 x - 24 - 12 z + 12 = 0 ⟹ 12 x - 12 z - 12 = 0 ⟹ x - z - 1 = 0

.Testing point

(1, 0, 0)

:

1 - 0 - 1 = 0

. (True)Point

(1, 0, 0)

lies on the plane.

Consider a plane passing through the points $A (2, 2, 1)$ , $B (3, 4, 2)$ and $C (7, 0, 6)$ . Which one of the following points lies on this plane?