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Mathematical Expression Editor

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We define the dot product and prove its algebraic properties.

VEC-0050: Dot Product and its Properties

Let and be vectors in . The dot product of and , denoted by , is given by

Find if and .

Note that the dot product of two vectors is a scalar. For this reason, the dot product
is sometimes called a scalar product.

Properties of the Dot Product

A quick examination of Example ex:dotex will convince you that the dot product is
commutative. In other words, . This and other properties of the dot product are
stated below.

The following properties hold for vectors , and in and scalar .

(a)

(b)

(c)

(d)

(e)

, and if and only if .

(f)

We will prove Property item:distributive. The remaining properties are left as exercises.