Chapter 6

Vector Products

6.1 The Dot ProductThere 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 6.1.1 Let a,b be two vectors in Rn define a ·b as

a ·b≡n

∑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 6.1.2 The dot product satisfies the following properties.

a ·b= b ·a (6.1)

a ·a≥ 0 and equals zero if and only if a= 0 (6.2)

(αa+βb) · c= α (a ·c)+β (b ·c) (6.3)

c · (αa+βb) = α (c ·a)+β (c ·b) (6.4)

|a|2 = a ·a (6.5)

You should verify these properties. Also be sure you understand that 6.4 follows fromthe first three and is therefore redundant. It is listed here for the sake of convenience.

Example 6.1.3 Find (1,2,0,−1) · (0,1,2,3) .

This equals 0+2+0+−3 =−1.

Example 6.1.4 Find the magnitude of a= (2,1,4,2) . That is, find |a| .

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