Dot Product and its Properties

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.

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

Proof of Property item:distributive:

We will illustrate Property item:norm with an example.

If we take the square root of both sides of the equation in Property item:norm, we get an alternative way to think of the length of a vector.

Practice Problems

Find the dot product of and if Answer:
Find the dot product of and if Answer:

Use vector to illustrate Property item:norm of Theorem th:dotproductproperties.
From the given list of vector pairs, identify ALL pairs of vectors that lie on perpendicular lines.
You may want to draw a picture and think about what you know about slopes of perpendicular lines.
, , , ,
Compute for each pair. What do you observe?
For each problem below
(a)
Find the value of that will make vectors and perpendicular.
Think of the -component as the “run” and the -component as the “rise”, then use what you know about slopes of perpendicular lines.
(b)
Find .

Answer:

Answer:
Answer:
(a)
Vector that lies on the line , has the form . Assuming that , find the general form for a vector that lies on a line perpendicular to .
What do you know about the slopes of perpendicular lines?
(b)
Find .
(c)
Formulate a conjecture about the dot product of perpendicular vectors.