Orthogonal Vector via Vector Cross Product

Vector Orthogonal to a Plane via the Cross Product:

If you find this page helpful and would recommend that I create more pages like this one, please let me know: Email to John Taylor

Find a vector orthogonal to the plane defined by

A(1, -3, 2), B(3, 4, -1), C(-2, 5, 3).

We can determine two vectors, AB, and AC in the plane. By taking their cross product, we can obtain a vector orthogonal to their plane.

First determine the vectors from the given points:

AB = [3 - 1]i + [4 - (-3)]j + [(-1) - 2]k = 2i + 7j - 3k.

AC = [-2 - 1]i + [5 - (-3)]j + [3 - 2] k = -3i + 8j + k.

Now form the cross product, which will be the desired orthogonal vector: