Math 144
Activity #9
Introduction to Vectors
Often times you hear people use the words speed and velocity. Is there a difference between the
two? If so, what is the difference? Discuss this with your group.
The two quantities are in fact different. Speed is considered a scalar quantity while velocity is a
vector. What is a vector? In this activity you will explore what a vector is and what are they used
for.
Quantities like area, volume, time, and speed are all considered to be scalar because they can be
assigned a single numerical value with a unit of measure. Force, momentum, displacement, and
velocity are some examples of vectors since they require more than one quantity to describe their
attributes. Each vector quantity needs both magnitude and direction in order to describe them
completely.
Let’s start with a geometric approach to vectors. Look at the three figures below. What do the
pictures represent? Now using what you remember from geometry, name each of the figures below
mathematically.
A
●
●
B
●
B
●
A
●
B
●
A
The figure that it is going to be explored further is the one in the middle. You may have
remembered it as being called a ray, but it can also be called a directed segment. It has a length, or
magnitude, that can be measured and an arrow to indicate a direction. For this reason, it can be a
referred to as a vector.
Vectors can be named using the same notation that is used for rays or using a bold, lower case letter
like v and u. Since it can be hard to write in bold using a pencil and paper, you will also see the
lower case letter with an arrow over the top. Name each of the vectors below.
B
D
A
C
F
A
G
H
E
What does it mean for quantities to be equal? Does this apply to vectors? If so, give a definition for
equivalent vectors?
Look back at the picture of the vectors on the first page; are there any vectors that are equal? If so,
what are they?
Consider the pictures of the vectors below, describe the relationship between them. Write a
mathematical equation to show the relationship.
v
u
When working with several vectors at a time, it can be easier if they are placed in a coordinate
plane. The graphical representation will help with analyzing the interaction between the vectors.
In the coordinate plane below, there are two vectors. Are the two vectors equal? Explain why or
why not.
How many units vertically are there between
the initial and terminal points for each
vector?
v
u
u
How many units horizontally are there
between the initial and terminal points for
each vector?
Are they the same?
What are the lengths of each vector?
Vectors v and u are in fact equivalent and the location of the vector is unimportant. For this reason,
a unique and equivalent vector can be placed anywhere in the coordinate plane, this is called the
position vector. What point would make a good place to put the initial point of the vector?
Draw the position vector for vector w.
Does the position vector look like something
that you have seen before in class? If so, what?
w
As mentioned before vectors have two components, a horizontal component and a vertical
component. On the graph below there is a vector drawn whose initial point is at the origin and
terminal point is at an arbitrary point, say a, b , in quadrant I. If we call this vector u, then another
notation could be u = a, b .
y
Draw the horizontal component, as a vector, for vector u.
Draw the vertical component, as a vector, for vector u.
What shape have you created?
x
What is the magnitude of the horizontal component?
What is the magnitude of the vertical component?
Write a formula to calculate the magnitude of vector u, denoted as |u|? Explain how you know this
will work.
You should have a picture that looks like the figure below, a right triangle drawn in quadrant one.
y
If this is not what your picture looks like go back, figure
out what you did wrong and fix it. Be sure to ask for help
if you need it.
b
x
a
You have worked with right triangles all semester long, so you should be able to find the angle 
that is in standard position. What is it? Is there another way to find  ? Write it down as well. Make
sure that every member of your group understands how and why each of your answers is correct.
Find the magnitude for each of the vectors drawn below. Then find the angle  , in standard
position, for each position vector.
w
v
Let’s go back to the position vector u = a, b with the vertical and horizontal components.
y
Since both the horizontal and vertical components are
represented as vectors, write each of them as such.
Horizontal component as a vector =
,
Vertical component as a vector =
,
x
Each component now has two components of their own, but in each case one of the components has
a value of zero. When the two components are put tip to tail, you can see they are connected by the
position vector that was started with. What do you get if you add the corresponding components
together?
If you did this correctly, you should have what is called the resultant vector; which is equivalent to
vector u drawn above. Placing vectors tip to tail in this fashion, will create a resultant vector. What
mathematical operation is this process equivalent to?
Find the resultant vector when vectors v and u
are placed tip to tail.
Does it matter which vector stays at the origin?
Explain or show how you came up with your
answer.
u
v
Draw each of the given vectors and then find the resultant vector for the sum of each.
1. v = 2,4 , w =  1,3
2. v = 0,6 , u = 5,2
If you can add vectors together, can you subtract them? Try it using the vectors w =  3,2 and
v = 5,4 . What are some of the things that you have to be careful of? If you are not sure ask your
instructor.