1. Vectors and Dot Products
6.4
Vectors and Dot Products
The Definition of the Dot Product of Two Vectors
The dot product of u = and v = is
Ex.’s Find each dot product.

2. Properties of the Dot Product
Let u, v, and w be vectors in the plane or in space and let c be a
scalar.

3. Let
Find
First, find u . v
Find u . 2v
= 2(u . v)
= 2(-14) = -28

4. The Angle Between Two Vectors
If is the angle between two nonzero vectors u and v, then
Find the angle between

5. Definition of Orthogonal Vectors (90 degree angles)
The vectors u and v are orthogonal if u . v = 0
Are the vectors orthogonal?
Find the dot product of the two vectors.
Because the dot product is 0, the two vectors are orthogonal.
End of notes.

6. Finding Vector Components
Let u and v be nonzero vectors such that u = w1 + w2
where w1 and w2 are orthogonal and w1 is parallel to
(or a scalar multiple of) v. The vectors w1 and w2 are
called vector components of u. The vector w1 is the
projection of u onto v and is denoted by w1 = projvu.
The vector w2 is given by w2 = u - w1.
is obtuse
is acute u u w2 w2 v v w1 w1

7. Projection of u onto v
Let u and v be nonzero vectors. The projection of u onto v is

8. Find the projection of onto Then write u
as the the sum of two orthogonal vectors, one which is projvu.
w1 = projvu =
w2 = u - w1 =
So,
u = w1 + w2 =