Question

The vector r = xˆi + y ˆj + zkˆ, called the position vector points from the origin (0, 0, 0) to an arbitrary point in space with coordinates (x, y, z). Use what you know about vectors to prove the following: All points (x, y, z) that satisfy the equation Ax + By + Cz = 0, where A, B, and C are constants, lie in a plane that passes through the origin and that is perpendicular to the vector Aˆi + Bˆj + Ckˆ. Sketch this vector and the plane.

Answer 1

The vectors r and p = Aˆi + Bˆj + Ckˆ are perpendicular between them. Thus, the plane equation come from the fact that the dot product is equal to zero.

Explanation:

The dot product of r and p

r*p = (xˆi + y ˆj + zk)*(Aˆi + Bˆj + Ckˆ) = Ax + By + Cz = 0