Write the equation for the plane.
-The plane through the points P(5, 2, -30) , Q(-3, 6, -26) and R(-1, -3, 29).
A) 4x + 7y + z = -4
B) 7x + y + 4z = 4
C) 4x + 7y + z = 4
D) 7x + y + 4z = -4

Question

Quizplus · Accepted Answer

To find the equation of the plane we need to first find two vectors that lie on the plane by taking the cross product of two vectors formed by the given points. 
Let PQ = (5 - (-3), 2 - 6, -30 - (-26)) = (8, -4, -4) and PR = (-1 - 5, -3 - 2, 29 - (-30)) = (-6, -5, 59). 
Then, the normal vector of the plane is PQ x PR = (221, 232, 32). 
To get the equation of the plane we use the point-normal form of the equation of a plane: 
a(x - x1) + b(y - y1) + c(z - z1) = 0 
where (a, b, c) is the normal vector and (x1, y1, z1) is a point on the plane. 
Using point P, we get: 
221(x - 5) + 232(y - 2) + 32(z + 30) = 0 
which simplifies to: 
4x + 7y + z = 4. Hence the answer is C.

Write the Equation for the Plane