Note that if and are real and orthogonal, the cosine of the angle between them is zero. In plane geometry (), the angle between two perpendicular lines is , and , as expected. More generally, orthogonality corresponds to the fact that two vectors in -space intersect at a right angle and are thus perpendicular geometrically.
Let and , as shown in Fig.5.8.
The inner product is . This shows that the vectors are orthogonal. As marked in the figure, the lines intersect at a right angle and are therefore perpendicular.
The Pythagorean Theorem in N-Space