Plane P is defined by A(2,0,0), B(0,4,0), C(0,0,2)

Question: Check if point D(-3,2,4) is in the plane P

Answer:

Lets Assume A to be origin in local coordinate system

Vector BA = B - A (column wise subtraction) = <-2, 4, 0>

Vector CA = C - A (column wise subtraction) = <-2, 0, 2>

Vector DA = D - A (column wise subtraction) = <-5, 2, 4>

Check if any vector is a linear combination of other two vectors:

Determinant[-2, -2, -5; -4, 0, 2; 0, 2, 4] = 0

Hence the point lines in the plane P

Check the Image