Section6.1Dot Products and Orthogonality¶ permalink

Objectives

Understand the relationship between the dot product, length, and distance.

Understand the relationship between the dot product and orthogonality.

Vocabulary words:dot product, length, distance, unit vector, unit vector in the direction of .

Essential vocabulary word:orthogonal.

In this chapter, it will be necessary to find the closest point on a subspace to a given point, like so:

The closest point has the property that the difference between the two points is orthogonal, or perpendicular, to the subspace. For this reason, we need to develop notions of orthogonality, length, and distance.

Subsection6.1.1The Dot Product

The basic construction in this section is the dot product, which measures angles between vectors and computes the length of a vector.

Definition

The dot product of two vectors in is

Thinking of as column vectors, this is the same as

For example,

Notice that the dot product of two vectors is a scalar.

You can do arithmetic with dot products mostly as usual, as long as you remember you can only dot two vectors together, and that the result is a scalar.

Properties of the Dot Product

Let be vectors in and let be a scalar.

Commutativity:

Distributivity with addition:

Distributivity with scalar multiplication:

The dot product of a vector with itself is an important special case:

Therefore, for any vector we have:

This leads to a good definition of length.

Fact

The length of a vector in is the number

It is easy to see why this is true for vectors in by the Pythagorean theorem.

For vectors in one can check that really is the length of although now this requires two applications of the Pythagorean theorem.

Note that the length of a vector is the length of the arrow; if we think in terms of points, then the length is its distance from the origin.

This says that scaling a vector by scales its length by For example,

Now that we have a good notion of length, we can define the distance between points in Recall that the difference between two points is naturally a vector, namely, the vector pointing from to

Definition

The distance between two points in is the length of the vector from to

In this section, we show how the dot product can be used to define orthogonality, i.e., when two vectors are perpendicular to each other.

Definition

Two vectors in are orthogonal or perpendicular if

Notation: means

Since for any vector the zero vector is orthogonal to every vector in

We motivate the above definition using the law of cosines in In our language, the law of cosines asserts that if are two nonzero vectors, and if is the angle between them, then

In particular, if and only if which happens if and only if Therefore,