A vector has magnitude and direction. There is an algebra and geometry of vectors which makes addition, subtraction, and scaling well-defined. The scalar or dot product of vectors measures the angle between them, in a way. It's useful to show if two vectors are perpendicular or parallel.

1. Section 9.2–3
Vectors and the Dot Product
Math 21a
February 6, 2008
Announcements
The MQC is open: Sun-Thu 8:30pm–10:30pm, SC 222
Homework for Friday 2/8:
Section 9.2: 4, 6, 26, 33, 34
Section 9.3: 10, 18, 24, 25, 34
Section 9.4: 1*

2. Outline
Vectors
Algebra of Vectors
Components
Standard basis vectors
Length
The Dot Product
Work
Concept
Properties
Uses

3. What is a vector?
Definition
A vector is something that has magnitude and direction
We denote vectors by boldface (v) or little arrows (v ). One is
good for print, one for script
Given two points A and B in flatland or spaceland, the vector
which starts at A and ends at B is called the displacement
−→
vector AB.
Two vectors are equal if they have the same magnitude and
direction (they need not overlap)
B D
v u
A C

4. Vector or scalar?
Definition
A scalar is another name for a real number.
Example
Which of these are vectors or scalars?
(i) Cost of a theater ticket
(ii) The current in a river
(iii) The initial flight path from Boston to New York
(iv) The population of the world

5. Vector or scalar?
Definition
A scalar is another name for a real number.
Example
Which of these are vectors or scalars?
(i) Cost of a theater ticket scalar
(ii) The current in a river vector
(iii) The initial flight path from Boston to New York vector
(iv) The population of the world scalar

6. Vector addition
Definition
If u and v are vectors positioned so the initial point of v is the
terminal point of u, the sum u + v is the vector whose initial point
is the initial point of u and whose terminal point is the terminal
point of v.
u
v v
u+v u+v
v
u u
The triangle law The parallelogram law

7. Opposite and difference
Definition
Given vectors u and v,
the opposite of v is the vector −v that has the same length
as v but points in the opposite direction
the difference u − v is the sum u + (−v)
v
u
−v
u−v

8. Scaling vectors
Definition
If c is a nonzero scalar and v is a vector, the scalar multiple cv is
the vector whose
length is |c| times the length of v
direction is the same as v if c > 0 and opposite v if c < 0
If c = 0, cv = 0.
v

9. Scaling vectors
Definition
If c is a nonzero scalar and v is a vector, the scalar multiple cv is
the vector whose
length is |c| times the length of v
direction is the same as v if c > 0 and opposite v if c < 0
If c = 0, cv = 0.
2v
v

10. Scaling vectors
Definition
If c is a nonzero scalar and v is a vector, the scalar multiple cv is
the vector whose
length is |c| times the length of v
direction is the same as v if c > 0 and opposite v if c < 0
If c = 0, cv = 0.
v
1
−2v

11. Properties
Theorem
Given vectors a, b, and c and scalars c and d, we have
1. a + b = b + a 5. c(a + b) = ca + cb
2. a + (b + c) = (a + b) + c 6. (c + d)a = ca + da
3. a + 0 = a 7. (cd)a = c(da)
4. a + (−a) = 0 8. 1a = a
These can be verified geometrically.

12. Components defined
Definition
Given a vector a, it’s often useful to move the tail to O and
measure the coordinates of the head. These are called the
components of a, and we write them like this:
a = a1 , a2 , a3
or just two components if the vector is the plane. Note the
angle brackets!

13. Components defined
Definition
Given a vector a, it’s often useful to move the tail to O and
measure the coordinates of the head. These are called the
components of a, and we write them like this:
a = a1 , a2 , a3
or just two components if the vector is the plane. Note the
angle brackets!
Given a point P in the plane or space, the position vector of
−→
P is the vector OP.

14. Components defined
Definition
Given a vector a, it’s often useful to move the tail to O and
measure the coordinates of the head. These are called the
components of a, and we write them like this:
a = a1 , a2 , a3
or just two components if the vector is the plane. Note the
angle brackets!
Given a point P in the plane or space, the position vector of
−→
P is the vector OP.
Fact −→
Given points A(x1 , y1 , z1 ) and B(x2 , y2 , z2 ) in space, the vector AB
has components
−→
AB = x2 − x1 , y2 − y1 , z2 − z1

15. Vector algebra in components
Theorem
If a = a1 , a2 , a3 and b = b1 , b2 , b3 , and c is a scalar, then
a + b = a1 + b1 , a2 + b2 , a3 + b3
a − b = a1 − b1 , a2 − b2 , a3 − b3
ca = ca1 , ca2 , ca3

16. Useful vectors
Definition
We define the standard basis vectors i = 1, 0, 0 , j = 0, 1, 0 ,
ı ˆ ˆ
k = 0, 0, 1 . In script, they’re often written as ˆ, , k.

17. Useful vectors
Definition
We define the standard basis vectors i = 1, 0, 0 , j = 0, 1, 0 ,
ı ˆ ˆ
k = 0, 0, 1 . In script, they’re often written as ˆ, , k.
Fact
Any vector a can be written as a linear combination of the
standard basis vectors
a1 , a2 , a3 = a1 i + a2 j + a3 k.

18. Length
Definition
Given a vector v, its length is the distance between its initial and
terminal points.

19. Length
Definition
Given a vector v, its length is the distance between its initial and
terminal points.
Fact
The length of a vector is the square root of the sum of the squares
of its components:
| a1 , a2 , a3 | = 2 2 2
a1 + a2 + a3

20. Early vector users
Caspar Wessel (Norwegian and Danish, 1745–1818)
Jean Robert Argand (French 1768–1822),
Carl Friedrich Gauss (German, 1777–1855)
Sir William Rowan Hamilton (Irish, 1805–1865)

21. Outline
Vectors
Algebra of Vectors
Components
Standard basis vectors
Length
The Dot Product
Work
Concept
Properties
Uses

22. Definition
Work is the energy needed to move an object by a force.

23. Definition
Work is the energy needed to move an object by a force.
If the force is expressed as a vector F and the displacement a
vector D, the work is
W = |F| |D| cos θ
where θ is the angle between the vectors.
θ D
Work is |F| times this distance
F

24. Definition
If a and b are any two vectors in the plane or in space, the dot
product (or scalar product) between them is the quantity
a · b = |a| |b| cos θ,
where θ is the angle between them.

25. Definition
If a and b are any two vectors in the plane or in space, the dot
product (or scalar product) between them is the quantity
a · b = |a| |b| cos θ,
where θ is the angle between them.
Another way to say this is that a · b is |b| times the length of the
projection of a onto b.
a
a · b is |b| times this length
b

26. Geometric properties of the dot product
Fact
Two vectors are perpendicular or orthogonal if their dot
π
product is zero (i.e., cos θ = 90◦ = )
2

27. Geometric properties of the dot product
Fact
Two vectors are perpendicular or orthogonal if their dot
π
product is zero (i.e., cos θ = 90◦ = )
2
The law of cosines can be expressed as
|a + b|2 = |a|2 + |b|2 − 2 |a| |b| cos θ
= |a|2 + |b|2 − 2a · b

28. Geometric properties of the dot product
Fact
Two vectors are perpendicular or orthogonal if their dot
π
product is zero (i.e., cos θ = 90◦ = )
2
The law of cosines can be expressed as
|a + b|2 = |a|2 + |b|2 − 2 |a| |b| cos θ
= |a|2 + |b|2 − 2a · b
In components, if a = a1 , a2 , a3 and b = b1 , b2 , b3 , then
a · b = a1 b1 + a2 b2 + a3 b3

29. More geometric properties of the dot product
Fact
The angle between two nonzero vectors a and b is given by
a·b
cos θ = ,
|a| |b|
where θ is taken to be between 0 and π.
Fact
The angle between two nonzero vectors a and b is
acute if a · b > 0

30. More geometric properties of the dot product
Fact
The angle between two nonzero vectors a and b is given by
a·b
cos θ = ,
|a| |b|
where θ is taken to be between 0 and π.
Fact
The angle between two nonzero vectors a and b is
acute if a · b > 0
obtuse if a · b < 0

31. More geometric properties of the dot product
Fact
The angle between two nonzero vectors a and b is given by
a·b
cos θ = ,
|a| |b|
where θ is taken to be between 0 and π.
Fact
The angle between two nonzero vectors a and b is
acute if a · b > 0
obtuse if a · b < 0
right if a · b = 0;

32. More geometric properties of the dot product
Fact
The angle between two nonzero vectors a and b is given by
a·b
cos θ = ,
|a| |b|
where θ is taken to be between 0 and π.
Fact
The angle between two nonzero vectors a and b is
acute if a · b > 0
obtuse if a · b < 0
right if a · b = 0;

33. More geometric properties of the dot product
Fact
The angle between two nonzero vectors a and b is given by
a·b
cos θ = ,
|a| |b|
where θ is taken to be between 0 and π.
Fact
The angle between two nonzero vectors a and b is
acute if a · b > 0
obtuse if a · b < 0
right if a · b = 0;
The vectors are parallel if a · b = ± |a| |b|.
b is a positive multiple of a if a · b = |a| |b|

34. More geometric properties of the dot product
Fact
The angle between two nonzero vectors a and b is given by
a·b
cos θ = ,
|a| |b|
where θ is taken to be between 0 and π.
Fact
The angle between two nonzero vectors a and b is
acute if a · b > 0
obtuse if a · b < 0
right if a · b = 0;
The vectors are parallel if a · b = ± |a| |b|.
b is a positive multiple of a if a · b = |a| |b|
b is a negative multiple of a if a · b = − |a| |b|

35. Examples
Example
Find the sum of the following pairs of vectors geometrically and
algebraically.
(i) a = 3, −1 and b = −2, 4
(ii) a = 0, 1, 2 and b = 0, 0, −3
What is the angle between the two vectors in each case?

36. Examples
Example
Find the sum of the following pairs of vectors geometrically and
algebraically.
(i) a = 3, −1 and b = −2, 4
(ii) a = 0, 1, 2 and b = 0, 0, −3
What is the angle between the two vectors in each case?
Solution
√ √
(i) a + b = 1, 3 , |a| = 10, |b| = 20. So
a·b −6 − 4 −10 1 3π
cos θ = = √ √ = √ √ = − √ =⇒ θ =
|a| |b| 10 20 10 20 2 4
(ii) a + b = 0, 1, −1 , while
0+0−6 2
cos θ = √ √ = −√
5 9 5

37. Properties
Fact
If a, b and c are vectors are c is a scalar, then
1. a · a = |a|2 4. (ca) · b = c(a · b) = a · (cb)
2. a · b = b · a
3. a · (b + c) = a · b + a · c 5. 0 · a = 0

38. Example
The dot product can be used to measure how similar two vectors
are. Consider it a compatibility index. If two vectors point in
approximately the same direction, we get a positive dot product. If
two vectors are orthogonal, we get a zero dot product. If two
vectors point in approximately opposite directions, we get a
negative dot product.
Consider the following categories,
1. Football
2. Sushi
3. Classical music
Now create a vector in R3 rating your preference in each category
from −5 to 5, where −5 expresses extreme dislike and 5 expresses
adoration. Dot your vector with your neighbor’s.

39. Example
Fifi, a poodle, drags her owner along a sidewalk that is 200 meters
long. If Fifi exerts a force of two newtons on the leash, and the
leash is at an angle 45◦ from the ground, how much work does Fifi
do?