(c)2024 C. Karimianpour
MAT188 WEEK 1-2 Vectors, Sets, Lines and Planes
Chapter 0 Learning Standards
(1) I can describe sets using set builder notation. (Writing)
(2) I can perform algebraic arithmetic on vectors in a Euclidean vector space Rn (e.g., addition,
scalar multiplication, norm, and dot product). (Computation)
(3) I can represent vectors arithmetic geometrically for vectors in R2 and R3 . (Visual/Geometry)
(4) I can decide whether given vectors are parallel. (Computation)
(5) I can decide whether given vectors are perpendicular. (Computation)
(6) I can compute the angle between any two vectors. (Computation)
(7) I can represent a line in R2 and R3 in vector form. (Conceptual )
(8) I can represent a line in R2 and R3 in parametric form. (Conceptual )
(9) Given an algebraic description of a line in R2 or R3 , I can visualize the line. (Visual/Geometry)
(10) I can represent a plane in R3 given a normal vector and a point. (Conceptual )
(11) Given an algebraic description of a plane, I can visualize the plane. (Visual/Geometry)
Vocabulary: vector, vector addition and scalar multiplication, dot product, orthogonal vectors, parallel
vectors, line, plane, vector form, parametric form, direction vector, normal vector, sets.
Reading from the textbook: Appendix A up to cross product.
1
MAT188 WEEK 1-2 Vectors, Sets, Lines and Planes
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Introduce
What are vectors? You may be familiar with vectors from physics. Given two points P and Q in the
space, the arrow, which is a directed line segment, starting from P and ending at Q is an example of a
vector. This arrow represents the displacement from P to Q. Let’s denote this vector by P⃗Q
For instance, if P = (1, 1) and Q = (3, 2) in the xy-plane, P⃗Q can be
understood as describing a walk we take in the plane to get from P to
Q. We start at P and we move two units horizontally and one unit
⃗
vertically.
The vector PQencodes this instruction, and is denoted by
2
2
P⃗Q =
. Note that
is just an instruction that says “move two
1
1
units horizontally and one unit vertically”. If we start at P and follow
this instruction, we will land on Q.
We can allow ourselves to begin walking from any point in the plane, and our
endpoints will depend on where we start. There are infinitely many possible
starting points, hence there are infinitely many arrows in the Cartesian plane
(a.k.a xy-plane) that visually represent the same vector. Where
do we end
2
up if we start from the origin and take a ride with the vector
?
1
x
Definition. If we represent a vector ⃗x = 1 by an arrow starting from the origin, we say the vector ⃗x
x2
is in its standard
position. In this case, the end point of the arrow is on the point (x1 , x2 ). We say the
x
vector ⃗x = 1 is the position vector of the point x = (x1 , x2 ).
x2
You can generalize this idea by considering points in 3D space. Let
P = (p
and Q = (q1 , q2 , q3 ) be points in xyz-space. Then
1 , p2 , p3 ) 
q1 − p1
⃗

P Q = q2 − p2  encodes the steps, this time in three directions,
q3 − p3
that one should take to go from P to Q. Once again, we can choose
where to start, and endpoints will depend
on our starting point.

1
What is the endpoint of the vector 3, in standard position?
2
In the examples above, the vectors had two or three entries, encoding the steps needed in each direction
to reach the tip of the arrow starting from the tail. In general, a vector is a column with n entries, where
n is a fixed positive integer. The entries of a vector are called components.
MAT188 WEEK 1-2 Vectors, Sets, Lines and Planes
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We can define a vector using the language of matrices. A matrix is a rectangular
array of entries. For
1 2 π
the purposes of our course, these entries are often numbers. For instance M =
is a matrix
−1 0 4
with two rows and three columns. We say Mis a 2 × 3 matrix1. A column vector is a matrix with only
1

one column and multiple rows for instance 2 is a column vector. A row vector is a matrix with one row
3
and multiple columns for instance 1 2 3 π is a row vector. In this course, unless otherwise stated,
by a vector we mean a column vector.
The collection of all column vectors with n components is denoted by Rn ; we will refer to Rn as the
n-dimensional Euclidean vector space. That is
 
a1
 a2 

Rn = {
 ...  | a1 , · · · , an are in R}
an
a
For example, R is the set of all vectors of the form 1 .
a2
a
R2 = { 1 | a1 , a2 are in R}
a2
Points vs Vectors. Points and vectors are different concepts. A point is a fixed position whereas a
vector is an instruction that encodes a displacement from one point to another. However, we can associate
each vector to a point and vise versa. We demonstrate this association in R2 .
a
Let O = (0, 0) be the origin. Visualize each vector 1 in R2 in its
a2
standard position (starting from the origin O). We can associate
this vector to its end point P = (a1 , a2 ). Conversely, given a point
a
1
⃗ =
P = (a1 , a2 ), we can associate it to its position vector OP
.
a2
This way we build a correspondence between points and vectors.
Through this correspondence, we can identify R2 , the set of all
vectors with two components, with the set of all points on the
Cartesian plane (xy-plane). That is, it is safe to think of R2 as the
Cartesian plane! Similarly, R3 can be visualized as xyz-space by
identifying
each point (a1 , a2 , a3 ) in the space by its position vector
 
a1
a2 . In this course we will often use this identification between
a3
points and vectors to visualize sets of vectors as familiar geometrical
objects such as lines and planes.
 
a1
 a2 

Under this identification, we can identify 
 ...  with (a1 , a2 , · · · , an ) and hence think of the n-dimensional
2
an
Euclidean space as the set of points with n components as well as set of vectors with n-components.
1This is a naive definition for a matrix that we go with for now; soon in this course, we will learn that matrices are WAY
more than just an array of numbers.
MAT188 WEEK 1-2 Vectors, Sets, Lines and Planes
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Vector Algebra
We can add two vectors with the same number of components, and we can scale a vector by multiplying
all of its components by a scalar.
 
 
1
0
2.5
3
4


Example (Vector Addition) Consider ⃗v = 
⃗ =
 π  and w
 1  in R and the scalar k = −4 in R.
−2
−1
Then


1
 5.5 

⃗v + w
⃗ =
π + 1
−3


−4
 −10 

and k⃗v = 
−4π  .
8
Definition (Vector addition and scalar multiplication)
The sum of two vectors ⃗v and w
⃗ in Rn is defined "componentwise". Given ⃗v , w
⃗ ∈ Rn
    

v1
w1
v1 + w 1
 v2   w2   v2 + w2 
   
.
⃗v + w
⃗ =
..
 ...  +  ...  = 

.
vn
wn
vn + w n
The product of a scalar k and a vector ⃗v is defined componentwise as well. Given a vector ⃗v ∈ Rn
and a scalar k ∈ R
   
v1
kv1
 v2   kv2 
  
k⃗v = k 
 ...  =  ...  .
vn
kvn
For vectors in R2 , and R3 , we can think of adding two vectors as
walking along these two vectors, one after the other. This way of
thinking of vector addition is often referred to as head to tail rule.
The vector you get is equal to the diagonal of the parallelogram
determined by the two vectors you add.
Multiplying a vector in R2 , and R3 by a scalar can be visualized as scaling the vector. Multiplying by a
negative scalar will flip the direction. To visualize the subtraction of two vectors, it is handy to realize
that −⃗v = (−1)⃗v and that ⃗u − ⃗v = ⃗u + (−⃗v ).
MAT188 WEEK 1-2 Vectors, Sets, Lines and Planes
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Note that in R2 and R3 , when multiplying a vector ⃗v by a scalar k, the resulting vector k⃗v is parallel to
⃗v . Vectors are parallel exactly when one is a scalar multiple of the other. We utilize this observation to
generalize the notion of parallel vectors to vectors in Rn :
Definition (Parallel Vectors)
We say two vectors ⃗v and w
⃗ are parallel if one is a scalar multiple of the other. That is if ⃗v = k w
⃗
for some k ∈ R or w
⃗ = k⃗v for some k ∈ R.



−3
1


−2
 and ⃗v =  6  are parallel in R4 , because −3⃗v = w.
⃗
Example The vectors ⃗v = 
 −9 
3
−12
4

The rules of vector algebra on the right (Theorem A.2 in Appendix A of the textbook) ensure that vector
arithmetic behaves similarly to that of real numbers for the most part.
Be aware that in general, we can not multiply two vectors.
MAT188 WEEK 1-2 Vectors, Sets, Lines and Planes
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Solidify
So far, we have seen how to generalize the notion of vectors from the physical concept of displacement
in spaces we can visualize, like R2 and R3 , to a mathematical notation of an n × 1 matrix in Rn for
any positive integer n. We made this generalization by setting up definitions. For instance, we defined
a vector in Rn to be an n × 1 matrix, and we defined vector addition and scalar multiplication in Rn
component wise. We observed how this definition matches our visual and intuitive understanding of the
vectors in R2 and R3 . This process is called abstraction and is a fundamental process in mathematics.
We will continue this process throughout this course to create a language that allows us to extend our
geometrical and visual notions in R2 and R3 to higher dimensions.
For vectors in R2 and R3 , we have geometric intuition about geometric concepts such as their length,
or geometric relationships between vectors, for example parallel vectors, and perpendicular vectors. Our
intuition comes through our spatial conception of these terms. We will use the power of abstraction
through definitions to extend these notions to vectors in Rn .
Let’s begin with making sense of the notion of length for vectors in Rn .
x
Example Let ⃗x = 1 in R2 . The length or magnitude or norm of ⃗x, denoted by ∥⃗x∥ is the length of
x2
the arrow that represents ⃗x.
Use Pythagorean
∥⃗x∥ =
theorem to see that
√
0
1
∥ = 1.
For example ∥
∥ = 2 and ∥
1
1
p
x21 + x22 .
Motivated by our understanding of length in R2 , we generalize
the notion of length to all vectors.
Definition (Norm)
 
v1
 v2 
n

Let ⃗v = 
 ...  be in R . The norm or magnitude or length of ⃗v is denoted by ∥⃗v ∥ and is defined
vn
to be
q
v12 + v22 + · · · + vn2 .
A useful notion that allows generalizing geometrical concepts in a coherent way is the notion of the dot
product.
MAT188 WEEK 1-2 Vectors, Sets, Lines and Planes
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Definition (Dot Product)
Let ⃗v and w
⃗ be (row or column) vectors with components v1 , v2 , · · · , vn and w1 , w2 , · · · , wn respectively. The dot product of ⃗v and w
⃗ is a scalar denoted by ⃗v · w
⃗ and is defined as
⃗v · w
⃗ = v1 w1 + v2 w2 + · · · + vn wn .
Exercise Verify that ∥⃗v ∥ =
√
⃗v · ⃗v .
The dot product can be interpreted geometrically. One can show
that for vectors in R2 and R3 .
⃗v · w
⃗ = cos θ∥⃗v ∥∥w∥,
⃗
where θ is the angle between ⃗v and w.
⃗ Solving for θ, we can say
that the angle between ⃗v and w
⃗ is given by
⃗v · w
⃗
arccos(
).
∥⃗v ∥∥w∥
⃗
This expression makes sense 2 for vectors in Rn , and motivates the next definition.
Definition (Angle)
Given vectors ⃗v and w,
⃗ the angle between ⃗v and w
⃗ is defined to be
⃗v · w
⃗
arccos
.
∥⃗v ∥∥w∥
⃗
Now we can extend our intuition about perpendicular vectors to Rn .
Definition (Perpendicular)
We say two vectors ⃗v and w
⃗ are perpendicular or orthogonal if ⃗v · w
⃗ = 0.
Exercise
(1) Does the definition of orthogonal vectors match your intuition in R2 and R3 ?
(2) If ⃗v and w
⃗ are orthogonal, what is the angle between them? Is there an exception to your answer?
(3) Is there a vector in Rn that is orthogonal to all other vectors in Rn by definition?
Our next step is to revisit how we describe lines and planes in R2 and R3 , and make sense of the concept
of lines and later in the course (not this week!) planes in Rn . We will describe these geometrical objects
using the language of sets. To make this transition smooth, first, let’s review the concept of a set.
Sets A set is a collection of objects. The objects in a set are called elements or members of that set.
For instance, the collection of all Toronto-based sports teams is a set and the Blue Jays is a member of
this set, or the collection of all even numbers is a set and 2022 is a member of this set. Let S be a set.
We write a ∈ S to signify that the object a is an element of the set S and write b ∈
/ S to signify that
the object b is not an element of the set S. For example, suppose E denotes the set of all even numbers.
Then 4 ∈ E and 3 ∈
/ E. Since a set has no distinguishing feature other than its content, there is a unique
2Cauchy Schwartz inequality guarantees −1 ≤
⃗
v ·w
⃗
∥⃗
v ∥∥w∥
⃗ ≤ 1.
MAT188 WEEK 1-2 Vectors, Sets, Lines and Planes
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set containing no elements which is called the empty set and is denoted by ∅ or {}. Some other common
sets have specific names and notations:3
R: the set of all real numbers
Z: the set of all integers or whole numbers
Q: the set of all rational numbers
N: the set of all natural numbers
C: the set of all complex numbers
There are multiple ways of describing a set: we can describe the set in words by specifying its members.
If the set is finite and has a manageable number of elements, we can describe it by listing all its members,
separated by a comma, inside curly brackets. For instance, the set M containing the names of the
following four mathematicians: Gauss, Noether, Mirzakhani, and Viazovska can be described as M =
{Noether, Gauss, Mirzakhani, Viazovska}. Note that the order in which the elements are listed does not
matter. Oftentimes, listing all the elements is not an efficient, or even feasible, way of describing a set.
For instance, think about the set of all even integers. It is impossible to list them all. Instead, we can
write
E = {2k | k ∈ Z},
Which reads “E is the set of all 2k’s where k is an integer”. In other word, “E is the set of all integer
multiples of 2.” We can describe the same set using a different language and notation
E = {z ∈ Z | z = 2k, for some integer k}
Which reads “E is the set of all z in Z such that z is some integer multiple of two”. In other words, “E
is the set of all z in Z such that z is even!”.
Definition (Subset)
We say a set A is a subset of a set B, and write A ⊆ B, if all the elements of A are also in B. In
other words, A ⊆ B, if for every a ∈ A, a ∈ B.
Definition (Equality of Sets)
We say sets A and B are equal if A is a subset of B and B is a subset of A. That is A = B if
A ⊆ B and B ⊆ A.
Definition (Union and Intersection of Sets)
Let X be a set and A and B be subsets of X. The union of A and B is a set that contains all
elements of A and B, that is
A ∪ B = {x ∈ X | x ∈ A or x ∈ B}
The intersection of A and B is the set of all common elements between A and B that is
A ∩ B = {x ∈ X | x ∈ A and x ∈ B}.
Example Let A = {2, 5, 7, π} and B = {4, π, 5} be subsets of R. Then A ∪ B = {2, 5, 7, π, 4} and
A ∩ B = {5, π}.
3You are already familiar with some of these sets, like R and Z, you may be less familiar with some others like Q and C.
Don’t worry!
MAT188 WEEK 1-2 Vectors, Sets, Lines and Planes
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Expand
Line in Rn You know that the equation y = −2x represents a line ℓ on the Cartesian xy-plane. This
line passes through the origin and has slope −2. Let’s describe ℓ in set notation: the line ℓ represents the
set of all points (x, y) on the Cartesian plane, that satisfy the equation y = −2x. That is ℓ is a subset of
the set of all the points on the Cartesian plane. Using set builder notation
ℓ = {(x, y) | y = −2x}.
If the value of x is the scalar t, then the value of y is −2t. The
scalar t is a parameter that varies over R. So it is allowed to take
any scalar value so that all the points on the line ℓ can be obtained.
Using set builder notation we can write
l = {(t, −2t) | t ∈ R}
Since we are working with vectors, we’d like to have a description of this line as a set of vectors. Recall
that we can identify points on the Cartesian plane with their position vector. Every point is the tip of
its position vector in R2 in standard position. This way, ℓ can be thought of as:
x
t
1
ℓ={
| y = −2x} = {
| t ∈ R} = {t
| t ∈ R},
y
−2t
−2
which is a subset of R2 . Hence we can write ℓ ⊂ R2 .
Another look at the latter description
of ℓ shows that ℓ is the set of
1
all scalar multiples of the vector
and this vector determines
−2
1
⃗
is a direction vector of ℓ.
the direction of ℓ. We say d =
−2
Indeed, any non-zero vector on the line can be a direction vector.
MAT188 WEEK 1-2 Vectors, Sets, Lines and Planes
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More generally, let ℓ be any line in R2 or R3 . We can identify points
on ℓ with their position vectors, and describe ℓ as the set of all such
position vectors. Let’s pick an arbitrary line and write it in set
builder notation.
Fix a nonzero vector d⃗ on the line. To do so, you will need two
⃗ The vector
points on the line serving as the tail and the tip of d.
d⃗ is a direction vector of the line. Pick the position vector p⃗ of
a point, any point, on the line. Then the position vector of an
⃗ for the appropriate choice
arbitrary point on ℓ is given by p⃗ + td,
of the parameter t. That is
ℓ = {⃗p + td⃗ | t ∈ R}
We often refer to ⃗v = p⃗ + td⃗ as the vector form or the vectorparametric form of the line ℓ. When we express a line in its vector
form, we always think of vectors in their standard position.
If we express d⃗ and p⃗ in terms of its components, we get the 
so-called
parametric

  equations of the line ℓ.
d1
p1
3
⃗



For instance, if ℓ is a line in R with a direction vector d = d2 . Let p⃗ = p2  be the position vector
d3
p3
of a point on the line. Then the vector form ⃗v = td⃗ + p⃗ yields the parametric equations:
(-1.1)
v1 = td1 + p1
v2 = td2 + p2
v3 = td3 + p3 .
Motivated by the vector form of a line in R2 and R3 , we define a line in Rn as follows:
Definition (Line)
A line in Rn is any set of the form
L = {tm
⃗ + ⃗b | t ∈ R},
where ⃗b and nonzero m
⃗ are fixed vectors in Rn .
MAT188 WEEK 1-2 Vectors, Sets, Lines and Planes
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Plane in Rn The only plane in R2 is the Cartesian plane R2 itself. So there is not much there to look at.
Let’s think about a plane P in R3 . First, identify all the points on the plane with their position vectors.
This way we can think of the plane P as a set of vectors in R3 whose tips, when in standard position, lie
on the plane. Next, we would like to describe this set using set builder notation.
Let ⃗n be a non-zero vector perpendicular to the plane P. We call ⃗n a normal vector of P. Suppose ⃗n is
drawn with the tail right on P. Because ⃗n is perpendicular to P, any vector lying on P is perpendicular
to ⃗n. Conversely, any vector that is perpendicular to ⃗n can be translated to lie on P.
Fix a point on P, and call its position vector p⃗. Now if ⃗x represents
the position vector of an arbitrary point on P, the vector ⃗x − p⃗ lies
on P (see the figure), and hence is perpendicular to ⃗n. That is
⃗n · (⃗x − p⃗) = 0
Indeed, we can describe P as {⃗x ∈ R3 | ⃗n · (⃗x − p⃗) = 0}, where p⃗
is the position vector of a specific point on P, and ⃗n is a normal
vector of P.
If we think of ⃗n, p⃗ and ⃗x in terms of their components we get the general form of the equation of P.
Example Let’s write the
form of the equation of a plane P that passes through the origin
 and

 general
x1
1
is perpendicular to ⃗n = 2. Since P passes through the origin, we can take p⃗ = ⃗0. Let ⃗x = x2  be
x3
3
the position vector of an arbitrary point on P. Then the normal equation ⃗x · ⃗n = 0 unpacks into
1x1 + 2x2 + 3x3 = 0.
In set builder notation we can write
 
x1
P = {x2  ∈ R3 | x1 + 2x2 + 3x3 = 0}
x3
 
n1
Similarly, we can show that the general form of the equation of a plane with normal vector ⃗n = n2 
n3
that passes through the point p = (p1 , p2 , p3 ) is
n1 x1 + n2 x2 + n3 x3 = D,
where D = n1 p1 + n2 p2 + n3 p3 .
Similar to lines in R3 , we can describe a plane in R3 in vector form. The vector form description is the
key that allows us to generalize and define a plane in Rn for any positive integer n. We mention the
vector form of a plane in Rn here, however, to understand why this expression indeed describes a plane,
we need to understand the concept of linear combination. We will see this concept next week!
Definition (Plane)
A plane in Rn is a set of the form
P = {tm
⃗ + s⃗n + ⃗b | s, t ∈ R},
where nonzero m,
⃗ nonzero ⃗n, and ⃗b are fixed vectors in Rn . Moreover, ⃗n, and m
⃗ are not parallel.
MAT188 WEEK 1-2 Vectors, Sets, Lines and Planes
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List of Definitions
Chapter 0 Definitions
Definition 0.1 (Set) A set is a collection of objects, called elements of the set.
Definition 0.2 (Subset) We say a set A is a subset of a set B, and write A ⊆ B, if all the elements of
A are also in B. In other words, A ⊆ B, if for every a ∈ A, a ∈ B.
Definition 0.3 (Equality of Sets) We say sets A and B are equal if A is a subset of B and B is a subset
of A. That is A = B if A ⊆ B and B ⊆ A.
Definition 0.4 (Union and Intersection of Sets) Let X be a set and A and B be subsets of X. The
union of A and B is a set that contains all elements of A and B, that is
A ∪ B = {x ∈ X | x ∈ A or x ∈ B}
The intersection of A and B is the set of all common elements between A and B that is
A ∩ B = {x ∈ X | x ∈ A and x ∈ B}.
 
v1

Definition 0.5 (Column Vector) A real column vector is a n×1 matrix ... , where vi ∈ R, 1 ≤ i ≤ n.
vn
Definition 0.6 (Row Vector) A real row vector is a 1 × n matrix v1 v2 · · · vn , where vi ∈ R,
1 ≤ i ≤ n.
Definition 0.7 (Euclidean Real Vector Space) The collection of all column vectors with n components
is denoted by Rn ; we will refer to Rn as the n-dimensional Euclidean vector space. That is
 
a1

a2 

Rn = {
 ...  | a1 , · · · , an are in R}
an
 
0
 .. 
.
 
1
Definition 0.8 (Standard Vectors) The standard vector ⃗ei ∈ Rm is defined by ⃗ei =  , a unit vector
0
.
 .. 
0
th
with zeros in all entries except for a 1 in the i entry.
Definition 0.9 (Vector addition and scalar multiplication) The sum of two vectors ⃗v and w
⃗ in Rn is
n
defined "componentwise". Given ⃗v , w
⃗ ∈R
    

v1
w1
v1 + w1
 v2   w2   v2 + w2 
   
.
⃗v + w
⃗ =
..
 ...  +  ...  = 

.
vn
wn
vn + wn
MAT188 WEEK 1-2 Vectors, Sets, Lines and Planes
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The product of a scalar k and a vector ⃗v is defined componentwise as well. Given a vector ⃗v ∈ Rn and a
scalar k ∈ R
   
v1
kv1
 v2   kv2 
  
k⃗v = k 
 ...  =  ...  .
vn
kvn
Definition 0.10 (Dot Product) Let ⃗v and w
⃗ be (row or column) vectors with components v1 , v2 , · · · , vn
and w1 , w2 , · · · , wn respectively. The dot product of ⃗v and w
⃗ is a scalar denoted by ⃗v · w
⃗ and is defined
as
⃗v · w
⃗ = v1 w1 + v2 w2 + · · · + vn wn .
 
v1
 v2 
n

Definition 0.11 (Norm) Let ⃗v = 
 ...  be in R . The norm or magnitude or length of ⃗v is denoted
vn
by ∥⃗v ∥ and is defined to be
q
v12 + v22 + · · · + vn2 .
Definition 0.12 (Parallel Vectors) We say two vectors ⃗v and w
⃗ are parallel if one is a scalar multiple
of the other. That is if ⃗v = k w
⃗ for some k ∈ R or w
⃗ = k⃗v for some k ∈ R.
Definition 0.13 (Perpendicular) We say two vectors ⃗v and w
⃗ are perpendicular or orthogonal if
⃗v · w
⃗ = 0.
Definition 0.14 (Angle) Given vectors ⃗v and w,
⃗ the angle between ⃗v and w
⃗ is defined to be
⃗v · w
⃗
arccos
.
∥⃗v ∥∥w∥
⃗
Definition 0.15 (Line) A line in Rn is any set of the form
L = {tm
⃗ + ⃗b | t ∈ R},
where ⃗b and nonzero m
⃗ are fixed vectors in Rn .
Definition 0.16 (Plane) A plane in Rn is a set of the form
P = {tm
⃗ + s⃗n + ⃗b | s, t ∈ R},
where nonzero m,
⃗ nonzero ⃗n, and ⃗b are fixed vectors in Rn . Moreover, ⃗n, and m
⃗ are not parallel.