Find a vector 𝑣⎯⎯⎯ v ¯ that is perpendicular to the plane through the points 𝐴=(2,1,−1) A = ( 2 , 1 , − 1 ) , 𝐵=(−2,3,0) B = ( − 2 , 3 , 0 ) , and 𝐶=(−5,−3,−5) C = ( − 5 , − 3 , − 5 ) .

Question

Find a vector  that is perpendicular to the plane through the points

Answer 1

Any vector parallel to the the normal vector of the plane that passes through the points A , B, and C.

Vector AB = < -4, 2,1 >

Vector AC = < -7,-4,-4>

Then the normal vector to the plane is the cross product of AB and AC

n = AB Χ AC =< -4, -23, 30 >

The any vector which is a scalar product of vector n is perpendicular to the plane