How do I find a vector that is perpendicular to a plane?

Let P be the plane in space that intersects the x-axis at 1, the y-axis at -4, and the z-axis at -2. Find a vector v that is perpendicular to P.|||The intersection information can be interpreted as the plane passes through the points (1, 0, 0), (0, -4, 0) and (0, 0, -2).





Okay, lets call these points A, B, and C. The vectors AB and AC (initial end at A and terminal end at B or C) both lay in the plane P. These vectors are





AB = (-1, -4, 0) and AC = (-1, 0, -2).





The normal to the plane has to be perpendicular to both of these vectors. This just screams cross product! A vector perpendicular to the plane is





v = AB x AC = (8, -2, -4).





You can take any nonzero scalar multiple of this.