Dot Product

linear-algebra

Definition

Dot Product

Let be two vectors of vector space , being a scalar field.

The dot product is an inner product defined as the sum over the dimension

where the different operator symbols are just different notations.

Einstein Notation:

Angle Formula

Let be two non-zero vectors, and let be the angle between them, then:

Derivation

We derive formula by comparing two ways to compute .

First, for any vector , its squared length is the dot product of the vector with itself:

Set , then:

Expand the right-hand side. In coordinates:

Note that each expands to . Hence:

Given associativity and commutativity, group terms like:

which can be rewritten as:

Using the definitions of norm and dot product, this becomes

and

Substituting these back gives

For the usual dot product, , so the middle two terms combine:

Now use and . This gives

On the other hand, and form two sides of a triangle with included angle , so the law of cosines gives

These are two expressions for the same quantity, so

Cancel and from both sides, then divide by . We get

Rearranging gives

so the angle can be recovered from the dot product.

Similarity

The dot product can be viewed as a measurement of similarity of two vectors because it quantifies how much two vectors are aligned. A higher dot product indicates greater degree of alignment or similarity between the vectors.

If is the angle between non-zero vectors and , then

So:

• if , then and the dot product is large and positive
• if , then and the dot product is
• if , then and the dot product is negative

This is why the dot product detects whether two vectors point in similar, orthogonal, or opposite directions.

Projections

Projection onto a Unit Vector

Start with a unit vector . Then is the signed scalar projection of onto the direction of .

If is the angle between and , then

Since is a unit vector, , so

This is exactly the signed length of the part of that points along .

The corresponding vector projection is

For a general non-zero vector , first make it a unit vector:

Then

and the projection onto is

This is why coefficients in an orthonormal basis can be recovered with the dot product. If

and the basis vectors satisfy

then for a fixed ,

So the dot product with picks out exactly the -th coefficient.