A plane is illustrated by any three points that are in the plane.  If a plane consists of the points P = (1, 0,0) , Q = (1,1,1) and R = (2, -1, 3) find out a vector that is orthogonal to the plane.

Solution

The one way which we know to get an orthogonal vector is to take a cross product. Thus, if we could find two vectors which we knew were in the plane and took the cross product of these two vectors we know that the cross product would be orthogonal to both of the vectors.  Though, as both the vectors are in the plane the cross product would then as well be orthogonal to the plane.

Thus, we require two vectors that are in the plane. This is in which the points come into the problem. As all three points lie in the plane any vector among them must as well be in the plane. There are several ways to get two vectors in between these points.  We will make use of the following two,

The cross product of these types of two vectors will be orthogonal to the plane. Thus, let's find the cross product.

Thus, the vector 4i^→ + j^→ - k^→ will be orthogonal to the plane consisting of the three points.