Chapter 14

Vector Products14.1 The Dot Product

There are two ways of multiplying vectors which are of great importance in applications.The first of these is called the dot product, also called the scalar product and sometimesthe inner product.

Definition 14.1.1 Let a,b be two vectors in Rp define a ·b as

a ·b≡p

∑k=1

akbk.

With this definition, there are several important properties satisfied by the dot product.In the statement of these properties, α and β will denote scalars and a,b,c will denotevectors.

Proposition 14.1.2 The dot product satisfies the following properties.

a ·b= b ·a (14.1)

a ·a≥ 0 and equals zero if and only if a= 0 (14.2)

(αa+βb) · c= α (a ·c)+β (b ·c) (14.3)

c · (αa+βb) = α (c ·a)+β (c ·b) (14.4)

|a|2 = a ·a (14.5)

You should verify these properties. Also be sure you understand that 14.4 follows fromthe first three and is therefore redundant. It is listed here for the sake of convenience.

Example 14.1.3 Find (1,2,0,−1) · (0,1,2,3) .

This equals 0+2+0+−3 =−1.

Example 14.1.4 Find the magnitude of a= (2,1,4,2) . That is, find |a| .

This is√(2,1,4,2) · (2,1,4,2) = 5.

The dot product satisfies the CauchySchwarz inequality. It has already been provedbut here is another proof. This proof will be based only on the above axioms for the dotproduct.

Theorem 14.1.5 The dot product satisfies the inequality

|a ·b| ≤ |a| |b| . (14.6)

Furthermore equality is obtained if and only if one of a or b is a scalar multiple of theother.

285