Mathematics of Physics and Engineering

Inner Product 11

Uv — U± V Uv = U±

Fig. 1.2.2 Orthogonal Projection

It follows from the picture that ||ti„|| = ||u||.| cos#| and uv has the same
direction as v if and only if 0 < 9 < -n/2. Comparing this with (1.2.1) we
conclude that

Uv UV V (1.2.4)

The vector uv is called the orthogonal projection of u on v, and is

denoted by uj_; the number u • t>/||t;|| is called the component of u in the

direction of v; note that «/||«|| is a unit vector. The verb "to project" comes

from Latin "to through forward." Let us emphasize that the orthogonal

projection of a vector is also a vector.

Let us now use the idea of the orthogonal projection to establish the

PROPERTIES OF THE INNER PRODUCT.

Consider two non-zero vectors u and w and a unit vector v. Then

(u + w) • v is the projection of u + w on v. From Figure 1.2.3, we conclude

that (u + w) • v = u • v + w • v.

(u + w)±

Fig. 1.2.3 Orthogonal Projection of Two Vectors