Lecture 2 (Text Section 12.3)
Several diﬀerent products can be deﬁned for vectors. In
the previous lecture we deﬁned the product of a scalar with a
vector. We now deﬁne a new product in which one vector is
multiplied by a second vector.
The dot product of vector u and v, represented symboli-
cally as u · v, is deﬁned by
u · v =
where ▯ is the angle (0 ≤ ▯ ≤ ▯) between the vectors.
Note that the resulting product is a scalar.
Two vectors are said to be orthogonal if their dot prod-
uct is equal to zero. For nonzero vectors, this implies that the
angle between the vectors is
The zero vector is orthogonal to all other vectors. The deﬁnition of the dot product does not provide a conve-
nient means of computation for component-wise representa-
tions of vectors. This is addressed by the following Theorem:
Given two vectors u = ⟨u ;u1;u 2 a3d v = ⟨v ;v ;v1⟩,2 3
u · v = u 1 1 u v2 2u v :3 3
Example: Find the dot product of the vectors u = ⟨2;3;−5⟩
and v = ⟨0;7;4⟩.
Knowing the dot product between two vectors allows for a
simple computation of the angles between two vectors. From
the deﬁnition of the dot product, we obtain the following re-
▯ = Example: Find the angle between the vectors given in the
Note if u = ⟨u 1u ;2 ⟩3 then
u · u =
Example: Use the dot product to calculate the length of
u = ⟨−2;3;5⟩. Properties of the Dot Product
Suppose u, v, and w are vectors and let c be a scalar.
1. u · v =
2. c(u · v) =
3. u · (v + w) =
These properties can be readily proved using the component-
wise representation of vectors.
Example: Prove property 3 above. Orthogonal Projections
Given two nonzero vectors u and v we can express u as the
sum of two vectors, one of which is parallel to v and the other
which is orthogonal to v.
The component of u parallel to v is called the projection
of u onto v and is represented symbolically by proj uv the
length of the projection vector is r