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
Prove Properties item:commutative, item:distributive-again, item:scalar, item:positive and item:norm of Theorem th:dotproductproperties.
From the given list of vector pairs, identify ALL pairs of vectors that lie on
perpendicular lines. Compute for each pair. What do you observe?
Problems prob:perpvectors2a-prob:perpvectors2c
For each problem below
- (a)
- Find the value of that will make vectors and perpendicular. (Hint: 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 .
- (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 .
- (b)
- Find .
- (c)
- Formulate a conjecture about the dot product of perpendicular vectors.