Transcript 12.3 The Dot Product

Chapter 12 – Vectors and the
Geometry of Space
12.3 – The Dot Product
12.3 – The Dot Product
1
Definition – Dot Product
Note: The result is not a vector. It is a real
number, a scalar. Sometimes the dot product is
called the scalar product or inner product.
12.3 – The Dot Product
2
Example 1 – pg.806 # 8
Find a  b
a = 3i + 2j - k
b = 4i + 5k
12.3 – The Dot Product
3
Properties of the Dot Product
12.3 – The Dot Product
4
Theorem – Dot Product
The dot product can be given a geometric
interpretation in terms of the angle  between a
and b.
12.3 – The Dot Product
5
Applying Law of Cosines
We can apply the Law of Cosines to the
triangle OAB and get the following
formulas:
12.3 – The Dot Product
6
Corollary – Dot Product
12.3 – The Dot Product
7
Example 2 – pg. 806 # 18
Find the angle between the vectors. (First
find an exact expression then
approximate to the nearest degree.)
a = <4, 0, 2>
b = <2, -1, 0>
12.3 – The Dot Product
8
Orthogonal Vectors
Two nonzero a and b are called
perpendicular or orthogonal if the angles
between them is  = /2.
12.3 – The Dot Product
9
Hints

The dot product is a way of
measuring the extent to
which the vectors point in
the same direction.

If the dot product is
positive, then the vectors
point in the same direction.

If the dot product is 0, the
vectors are perpendicular.

If the dot product is
negative, the vectors point
in opposite directions.
12.3 – The Dot Product
10
Visualization

The Dot Product of Two Vectors
12.3 – The Dot Product
11
Example 3
For what values of b are the given
vectors orthogonal?
<-6, b, 2>
<b, b2, b>
12.3 – The Dot Product
12
Definition – Directional Angles
The directional angles of a nonzero
vector a are the angles , , and 
in the interval from 0 to pi that a
makes with the positive axes.
12.3 – The Dot Product
13
Definition – Direction Cosines
We get the direction cosines of a
vector a by taking the cosines of the
direction angles. We get the
following formulas
12.3 – The Dot Product
14
Continued
12.3 – The Dot Product
15
Example 4 pg. 806 #35

Find the direction cosines and
direction angles of the vector. Give
the direction angles correct to the
nearest degree.
i – 2j – 3k
12.3 – The Dot Product
16
Definition - Vector Projection

If S is the foot of the perpendicular from R
to the line containing PQ , then the vector
with representation PS is called the vector
projection of b onto a and is denoted by
projab. (think of it as a shadow of b.)
12.3 – The Dot Product
17
Definition continued
12.3 – The Dot Product
18
Visualization

Vector Projections
12.3 – The Dot Product
19
Definition – Scalar Projection
The scalar projection or component of b
onto a is defined to be the signed
magnitude of the vector projection, which
is the number |b|cos, where  is the
angle between a and b. This is denoted by
compab.
12.3 – The Dot Product
20
Definition continued
12.3 – The Dot Product
21
Example 5 – pg807 #42

Find the scalar and vector
projections of b onto a.
a = <-2, 3, -6>
b = <5, -1, 4>
12.3 – The Dot Product
22
More Examples
The video examples below are from
section 12.3 in your textbook. Please
watch them on your own time for
extra instruction. Each video is
about 2 minutes in length.
◦ Example 1
◦ Example 3
◦ Example 6
12.3 – The Dot Product
23
Demonstrations
Feel free to explore these
demonstrations below.
 The Dot Product
 Vectors in 3D
 Vector Projections
12.3 – The Dot Product
24