Chapter 3

Vector Products3.1 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 3.1.1 Let a,b be two vectors in Rn define a ·b as

a ·b≡n

∑k=1

akbk.

The dot product a ·b is sometimes denoted as (a,b) of ⟨a,b⟩ where a comma replaces ·.

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 3.1.2 The dot product satisfies the following properties.

a ·b = b ·a (3.1)

a ·a≥ 0 and equals zero if and only if a = 0 (3.2)

(αa+βb) · c =α (a · c)+β (b · c) (3.3)

c · (αa+βb) = α (c ·a)+β (c ·b) (3.4)

|a|2 = a ·a (3.5)

You should verify these properties. Also be sure you understand that 3.4 follows fromthe first three and is therefore redundant. It is listed here for the sake of convenience.

Example 3.1.3 Find (1,2,0,−1) · (0,1,2,3) .

This equals 0+2+0+−3 =−1.

Example 3.1.4 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 a fundamental inequality known as the Cauchy Schwarz in-equality.

Theorem 3.1.5 The dot product satisfies the inequality

|a ·b| ≤ |a| |b| . (3.6)

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

31