Chapter 12 – Vectors and the
Geometry of Space
12.3 – The Dot Product
12.3 – The Dot Product
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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.
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Example 1 – pg.806 # 8
Find a  b
a = 3i + 2j - k
b = 4i + 5k
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Properties of the Dot Product
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Theorem – Dot Product
The dot product can be given a geometric
interpretation in terms of the angle  between a
and b.
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Applying Law of Cosines
We can apply the Law of Cosines to the
triangle OAB and get the following
formulas:
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Corollary – Dot Product
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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>
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Orthogonal Vectors
Two nonzero a and b are called
perpendicular or orthogonal if the angles
between them is  = /2.
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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.
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Visualization

The Dot Product of Two Vectors
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Example 3
For what values of b are the given
vectors orthogonal?
<-6, b, 2>
<b, b2, b>
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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.
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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
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Continued
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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
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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.)
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Definition continued
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Visualization

Vector Projections
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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.
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Definition continued
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Example 5 – pg807 #42

Find the scalar and vector
projections of b onto a.
a = <-2, 3, -6>
b = <5, -1, 4>
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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
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Demonstrations
Feel free to explore these
demonstrations below.
 The Dot Product
 Vectors in 3D
 Vector Projections
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