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Dot Product: Equivalence of Definitions

We have operated with the dot product as an operation with two different definitions. In this section, we will show, in multiple different ways, that these definitions are equivalent.

Table of Contents​

The Definitions to Equate​

The dot product of two vectors and can be defined in two ways, as we have previously seen:

We will show that these two definitions are equivalent. That is:

By the Cosine Rule​

Brief Overview

This proof assumes the component definition of the dot product () and uses the cosine rule to derive the geometric definition of the dot product ().

Additionally, it uses the corollary that the square of the magnitude of a vector is the dot product of the vector with itself; .

For any triangle with sides , , and , and angles , , and , the cosine rule states:

We can apply this rule to the triangle formed by the vectors and , and their difference . Below is a diagram of the triangle:

If we let , , and , we can apply the cosine rule to the triangle:

Recall that the square of the magnitude of a vector is the dot product of the vector with itself:

Substituting this back into the cosine rule equation, we get:

Finally, we can cancel out the and terms to get the equivalence of the dot product definitions:

By Vector Decomposition​