Module 8 Lesson 1: What is a Vector?
Many of our application problems in mathematics involve manipulating a single quantity
such as time, temperature, or area. However, many quantities in mathematics and
physics (such as force and velocity) involve two quantities simultaneously: magnitude
and direction.
In general, a vector is a quantity that has a magnitude and a direction. The magnitude
refers to the length or size of the vector, and the direction refers to “which way” the
vector is pointing. See the image below.
Vectors can be represented as directed line segments that have an initial point P and a
terminal point Q. See the image above.
We have three ways to name vectors.
 With the initial point and terminal point, such as “ ⃑⃑⃑⃑⃑
𝑃𝑄 ”.
⃑ ”.
 Giving it a letter name, such as “ 𝑉
 Or by a boldface lowercase letter, such as u, v, or w.
In this course, we will most often denote vectors using their initial and terminal points
and by u, v, or w. Be familiar in working with both types of notation. If you happen to
be looking up vectors from other resources on the internet, you may find them denoted
differently.
In general, the magnitude of a vector is the length of the vector and can be denoted by
⃑⃑⃑⃑⃑ ‖ (depending on which type of notation is used).
‖𝐯‖ or ‖𝑃𝑄
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Ex. Finding the Magnitude (or Length) of a Vector
Let u be represented by the directed line segment from P = (0, 0) to Q = (3, 2), and let v
be represented by the directed line segment from R = (1, 2) to S = (4, 4). Find the
magnitude (or length) of u and the magnitude (or length) of v.
We can find the magnitude of each vector by using the distance formula.
⃑⃑⃑⃑⃑ ‖ = √(3 − 0)2 + (2 − 0)2 = √13
‖𝐮‖ = ‖𝑃𝑄
and
⃑⃑⃑⃑⃑ ‖ = √(4 − 1)2 + (4 − 2)2 = √13
‖𝐯‖ = ‖𝑅𝑆
Note: You can also find the magnitude by creating a right triangle and using the
Pythagorean Theorem. That’s all the distance formula is anyways—an application of the
Pythagorean Theorem. See how to find the magnitude of v below.
‖𝐯‖ = √𝟑𝟐 + 𝟐𝟐 = √𝟏𝟑
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Equal Vectors
Two vectors are equal if they have the same magnitude and direction.
In the previous example, the two vectors u and v have the same magnitude (they both
had a magnitude of √13). The direction of the two vectors is also the same as they are
both directed toward the upper right and both have a slope of 2/3.
Therefore, we can conclude that u = v.
The Component Form of a Vector
The component form of a vector v is written as
𝐯 = ⟨𝑣1 , 𝑣2 ⟩
We call the coordinates of 𝑣1 and 𝑣2 the components of v.
To find the component form of the vector, let the initial point 𝑃 = (𝑝1 , 𝑝2 )
and the terminal side 𝑄 = (𝑞1 , 𝑞2 ). Then we can write
⃑⃑⃑⃑⃑ = ⟨𝑞1 − 𝑝1 , 𝑞2 − 𝑝2 ⟩ = ⟨𝑣1 , 𝑣2 ⟩ = 𝐯
𝑃𝑄
Note: If both the
initial point and
the terminal
point lie at the
origin, v is the
zero vector and
is denoted by
𝟎 = ⟨0, 0⟩.
Ex. Finding the Component Form of a Vector
Find the component form of the vector v that has initial point (4, -7) and terminal point
(-1, 5).
Let’s let P = (4, -7) and Q = (-1, 5).
It might help to sketch this vector in the coordinate plane.
Therefore, v = ⟨(−1) − 4, 5 − (−7)⟩ = ⟨−5, 12⟩.
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Unit Vectors
In many applications of vectors, it is helpful to find the unit vector. When given a vector
v, the unit vector of v is a vector that has the same direction as v but has a length of 1.
To find the unit vector, divide the vector by it’s magnitude.
𝐯
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u = the unit vector = ‖𝐯‖ = (‖𝐯‖) 𝐯
Note: Sometimes this unit vector is called the unit vector in the direction of v. The
“original vector” and the unit vector always have the same direction, it is only the length
of the vector that changes.
Ex. Find the unit vector in the direction of 𝐯 = ⟨−4, 4⟩.
Let’s first find the magnitude of v. Since this vector is written in component
form, we know the horizontal component and vertical component. These values
are already the difference between the x-coordinates and y-coordinates of the
initial and terminal points. Therefore, we have
‖𝐯‖ = √(−4)2 + 42 = √32 = 4√2
Continuing, let’s use the formula above and simplify.
1
1
(‖𝐯‖) 𝐯 = 4√2 ⟨−4, 4⟩
−4
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= ⟨4√2 , 4√2⟩
Use scalar multiplication
= ⟨−
Simplify
=
1
,
1
⟩
√2 √2
√2 √2
⟨− , ⟩
2 2
Simplify/Rationalize
Conclusion
Those are the “basics” of vectors. In the following lessons, you will learn how to operate
with vectors using addition and multiplication, as well as how to apply vectors to real
world applications.
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