Lesson Explainer: Graphical Operations on Vectors
Mathematics • 12th Grade
In this explainer, we will learn how to do operations on vectors graphically
using triangle and parallelogram rules.
Vectors are objects that are entirely defined by their magnitude and direction. We recall that we can represent vectors by a directed line segment, in a suitable
space, where the length of the line segment tells us the magnitude of the vector
and the direction is indicated by the initial and terminal points of the directed
line segment. In this space, we can think of vectors as representing displacement,
from the initial point to the terminal point.
For example, the vector from to
can be represented as the directed line segment from to
.
The length of this line segment is the magnitude of
, written
, and the direction
is shown by the arrow. It is worth reiterating that a vector is defined entirely by
its magnitude and direction, so any two vectors with the same magnitude and
direction will be equal. In particular, we can draw the vector anywhere in the
-plane; this will not change its magnitude or direction,
and hence it will still be the same vector.
We can also represent this vector by the horizontal change and vertical change. If and
, then from the diagram
(or the coordinates) we can find the horizontal and vertical changes when moving
from to .
The change in the horizontal coordinate is
and the change in the vertical coordinate is . We write this as
. The first component tells us the horizontal displacement of the vector and the second
component tells us the vertical displacement of the vector.
This then allows us to add vectors together. In terms of displacement, the sum of
two vectors will be the total displacement of both vectors. In terms of the
components, we recall that this means if we have
and ,
then ; we add the
horizontal and vertical components separately. This then allows us to add vectors
together graphically.
For example, if we have the vector ,
then in this space, this is the displacement from point
to point . Similarly,
is the displacement from point
to point . Their sum,
, should be the total
displacement of both vectors, that is, the displacement from
to and then from to
. This is the vector ,
and we can show this graphically.
We can see that the displacement from to
is the same as the displacement from
to and then from
to , since their initial and terminal points
are the same. This is often referred to as the triangle rule for vectors. We can state this as follows.
Rule: Triangle Rule for Vectors
For any three points , , and
, we have
as shown in the diagram.
As a result of the triangle rule, if we have two vectors
and
represented graphically, then we can add these vectors together by sketching the
terminal point of to be the initial point of
, as shown.
Before we see how to apply this to problems involving vectors, there is one more
rule we can show. Consider parallelogram .
In a parallelogram, opposite sides are parallel and have the same length; therefore,
We can apply the triangle rule for vectors to this diagram by considering the
sides as vectors and adding vector to the diagram,
as shown.
Applying the triangle rule for vectors to points ,
, and , we have
We know that
since they have the same magnitude and direction. We can see this graphically as
we can translate the vectors on top of each other. Substituting this into the
equation gives
This is known as the parallelogram rule for vectors, and we can summarize this is
as follows.
Rule: Parallelogram Rule for Vectors
For vectors and
with the same initial point,
,
where is the point that makes
a parallelogram, as shown in the following diagram.
Since vectors have a number of uses, for example, as a tool to evaluate the
resultant of forces or to solve geometric problems, the triangle and parallelogram
rules can be thought of entirely in terms of the field we are working in. However, it is usually easier to convert the problem into vectors.
Letβs start by looking at some examples of adding vectors together
graphically.
Example 1: Finding the Components of the Sum of End-to-End Vectors on a
Diagram
Shown on the grid of unit squares are the vectors
and .
What are the components of ?
What are the components of ?
What are the components of ?
Answer
Part 1
We recall that the components of a vector represented graphically are the
horizontal and vertical displacements of the vector in this space. In particular,
for a vector from
to , its components will be the
difference in the coordinates:
There are a few ways of using this to determine the components of vector
. For example, we could introduce a coordinate
system onto the grid of unit squares. However, this is not necessary. Instead,
we only need to know the horizontal and vertical displacements when traveling
from the initial point of to the terminal point
of . Letβs start with the horizontal
displacement.
Moving from the tail to the tip of vector ,
we travel over three of the squares. Since these are unit squares and we
travel to the right, we can write this as displacement of
+3 units.
We can do the same vertically.
We move two units upward, so the displacement is +2 units. This gives us both components of our vector. Remember, we do not need to
include the positive symbol; this gives us
Part 2
We can follow the same procedure for vector .
First, we see that we move two units to the right, so the first component of the
vector will be 2. We need to be careful when checking the vertical displacement of
the vector; since we move downward, the value will be negative.
Traveling from the tail to the tip of vector ,
we move three units downward. Therefore, the vertical component of
will be . Hence,
Part 3
There are two methods we can use to add the vectors together.
First, we can recall that we add vectors together component-wise and then use
our answers in the first two parts of the question; this gives
However, this result relies on us finding the components of both vectors
correctly and then not making any mistakes in the calculation. This becomes more of
a problem the more vectors we are asked to add together. Instead, we can also add
the two vectors together graphically.
Since we add the components of each vector separately, we can add vectors together
using the βtip-to-tailβ method. We sketch the vectors so that the
tip of one vector is aligned with the tail of the next one, so the sum of the
vectors is then the vector from the initial point of the first vector to the
terminal point of the final vector. Since the vectors are already sketched in this
way, we have the following.
The vector
is represented by the third side in the triangle since the directions of
and match. We can then find the components of
using the diagram.
Following the vector ,
we move 5 units right and 1 unit down, so the horizontal component is 5 and the
vertical component is , giving us
We can see that this agrees with the direct calculation we did above.
Example 2: Finding the Sum of Two Vectors Given Graphically
Which graph represents
,
where
and ?
Answer
Since we are given the components of and
, we can add the vectors component-wise
to get
Looking at the options, we can then see that only option
A has ,
,
and , so it is the correct answer. The problem with answering the question in this manner is that we will
often be asked to add vectors together graphically without being given the
options, so we will also construct the diagram ourselves.
We can start by sketching vectors
and
onto a diagram, where we recall that the first component tells us the
horizontal displacement and the second component tells us the vertical
displacement. It is also worth noting that we can sketch the vectors anywhere
on the plane. However, for simplicity, we will start both vectors at the
origin. First,
will travel 3 units right and 4 units up to give us the following.
We can do the same with vector ,
which will travel 4 units right and 1 unit upward, as shown.
It is worth noting that since we started our vectors from the origin,
the coordinates of the terminal points will be equal to the corresponding
components of the vector; these are often called position vectors.
To add these two vectors together graphically, we start by sketching them
so that the terminal point of one vector is the initial point of the other. We will move to have its initial point at
; it will still move 4 units
right and 1 unit up giving us the following.
Finally, the triangle rule for vectors tells us that the sum of these two
vectors will be the vector having the initial point of
and the terminal point of
since these vectors are sketched tip to
tail, giving us the following.
We can see in the diagram that travels 7 units right and 5 units up, so
,
and this is shown in option A.
In our previous examples, we used the graphical representations alongside their
components to solve problems. In our next example, we will use only the graphical
representation of the vectors to solve a geometric problem.
Example 3: Identifying the Correct Diagonal in the Parallelogram Law
Which vector is equivalent to ?
Answer
Quadrilateral has opposite sides as equal vectors. Since equal vectors have the same magnitude and direction, we can conclude
that is a parallelogram. We want to apply the
parallelogram rule for vectors to add these vectors. To do this, we notice
that is the vector from to , so
Similarly,
The parallelogram rule for adding vectors then tells us that if
and
have the same initial point,
, then
where is the point that makes
a parallelogram. Vector
is the diagonal of the parallelogram as
shown.
Therefore, .
Using the above example, we can show a useful result from the parallelogram we
constructed.
By applying the triangle rule for vectors on the points ,
, and , we have
In terms of and
, this states that
In other words, this is a geometric demonstration of the commutativity of vector
addition.
Thus far, we have only considered adding to vectors together. We will now consider
an example where we are asked to find the difference of two vectors.
Example 4: Identifying the Correct Graphical Representation of the Difference of Two
Vectors
Which of the following parallelograms shows a valid way of obtaining
?
Answer
To subtract two vectors graphically, we will use two results. First, we can
rewrite the expression as the sum of two vectors:
This tells us the difference between and
is the same as the sum of
and ,
and we know how to add vectors together graphically by using both the triangle
and parallelogram rules. Since the options have parallelograms included,
we will use the parallelogram rule.
The parallelogram rule for vector addition tells us that if
and
are vectors with the same initial point,
, then
where is the point that makes
a parallelogram, as shown in the following diagram.
If we label the vectors, which are the sides of the parallelogram,
and
, we have the following.
Then, since , we can replace
the vector on the diagonal with ,
which is the same as option C.
We can then ask the following question: given two vectors
and , graphically,
how can we find ? To do this,
we will start by sketching and
to have the same initial point, say
; we will also label the terminal points of the vectors
and as shown.
We want to determine . Recall that
and that multiplying a vector by switches its direction and does
not change its magnitude. In our diagram, this means
is the vector from to , as shown.
We can then see that the terminal point of
is the initial point of , so we can add these
vectors by using the triangle rule for vector addition; we have the following.
We can then use commutativity of vector addition to write
. This means
We can then represent this graphically as shown.
Letβs now see a few examples of applying this reasoning to subtract two
vectors given graphically.
Example 5: Finding the Difference of Two Vectors Represented Graphically
Which of the following is equivalent to ?
Answer
We are asked to find the difference of two vectors. To do this, we start by
noting that multiplying a vector by switches its direction
but leaves the magnitude unchanged, so
Hence,
We can draw both of these vectors on the diagram.
Since the terminal point of is the initial
point of , we can add these vectors by using
the triangle rule for vector addition, which tells us
which we can show on the diagram.
However, this is not one of the given options. We need to also use the fact
that is a parallelogram where opposite sides have
equal size and are parallel. Therefore,
Hence, the answer is option E, .
Example 6: Solving a Geometric Problem Involving Vectors as the Sides
of a Parallelogram
Which of the following is equivalent to
?
Answer
We want to determine a vector equivalent to the given expression by using the
diagram. To do this, we first note that for a vector
,
will have the same direction but half the magnitude. So, we will start by
finding a vector equivalent to
. We can do this by first recalling that
, so
We can add these vectors to the diagram.
To add vectors together graphically, we can use the parallelogram rule for
vectors. Since the initial points of both vectors are the same, their sum
will be the diagonal of the parallelogram.
Hence,
We want to find of this vector, so we need a
vector in the same direction as
but with half the magnitude. Since is the midpoint of
, there are two options for this;
both and
will have the same direction as
but half the magnitude.
Hence,
So, the answer is option B, .
It is worth noting that we can apply these rules for vector addition multiple
times. For example, consider the following diagram.
If we wanted to find , we can do this by applying the triangle
rule for vector addition twice. First, we have
, as shown in the following diagram.
We can then add and
with the triangle rule for vector addition. We have , as shown.
Hence,
Graphically, we can think about this as the three vectors
, ,
and all being drawn tip to tail, so their sum is
the vector that has the initial point and the terminal point
. We can apply this same reasoning to even more complicated
shapes as we will see in our final example.
Example 7: Simplifying the Sum of Vectors That are the Sides of a Polygon
Complete: In the following figure,
.
Answer
We begin by adding all of the vectors in the sum onto the diagram.
There are two ways we can determine this sum: we can do this directly by
recalling that the sum of any number of vectors drawn tip to tail will have
the initial point of the first vector and the terminal point of the final
vector. On our diagram, this will be vector
as shown.
It might be easier to think of this as following the vectors from
all the way to . We have
which is option B.
To see why this is true, we need to apply the triangle rule, which states
that for any points , ,
and ,
We can apply this to the first two vectors in the sum to get
so
We can then apply the triangle rule again and see that
so
Finally, we apply the triangle rule one more time to get
and hence
Therefore, the answer is option B, .
Letβs finish by recapping some of the important points of this
explainer.
Key Points
The triangle rule for vectors tells us that, for any three points
, , and ,
The triangle rule for vectors allows us to add two vectors graphically by drawing
them such that the initial point of one vector is the terminal point of the other.
The parallelogram rule for vectors tells us that
where is the point that makes
a parallelogram.
The parallelogram rule for vectors allows us to add vectors together graphically by
drawing them such that they have the same initial point. Then, their resultant will
be given by the diagonal of the parallelogram spanned by the vectors.
We can subtract two vectors by using the fact that, for any three points
, , and ,
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