Video Transcript
In this lesson, we’re going to
learn how to subtract one vector from another vector, both graphically and using
unit vector notation.
We will start with the graphical
method. Before we begin to learn how to
subtract vectors graphically, let’s refresh our memory on what a vector is and how
to represent a vector graphically. A vector is a quantity that has
both magnitude and direction. Let’s draw an example of two
vectors 𝐀 and 𝐁. Let’s make vector 𝐀 have a
magnitude of five units and be oriented horizontally to the right of our screen. Note that when we label our vector
𝐀, we draw a half arrow over the top of the letter. This is a common convention to show
that a variable is a vector. In text form, it is common for the
variable to be bolded. We will see this form when we do
example problems at the end of our lesson.
We can draw vector 𝐁 to have the
same magnitude of five units, but we can orient it vertically to the top of our
screen. Both of these vectors have the same
magnitude but are heading in different directions. We need to take extra care and make
sure that we draw the vectors to the correct length in the representation because it
will affect the outcome when we subtract them. Before we move on to subtracting
our vectors, let’s draw one last vector that has components along both the
horizontal and vertical axes.
The graphical representation of
vector 𝐂 has a length of seven units along the horizontal to the right of our
screen and four units along the vertical to the top of our screen. We can see that vector 𝐂 has a
greater horizontal magnitude than vector 𝐀 but smaller vertical magnitude than
vector 𝐁. Now that we have done a quick recap
on what vectors are and how to draw them graphically, let’s dive into how to
subtract vectors graphically.
When we’re subtracting vector 𝐁
from vector 𝐀, we can write this expression as 𝐕, the resultant vector, is equal
to vector 𝐀 minus vector 𝐁. The approach we’ll be using to
subtract vectors graphically is to add the negative of vector 𝐁 to vector 𝐀 and
then apply the tip-to-tail method of adding vectors. We can do this because subtracting
vector 𝐁 from vector 𝐀 is the same thing as adding negative vector 𝐁 to vector
𝐀.
When we talk about the negative of
a vector, in this case, negative vector 𝐁, that means we need to rotate our vector
180 degrees from its starting position. Our new vector will be pointing in
the complete opposite direction both horizontally and vertically from where the
vector was originally. In the tip-to-tail method, one
vector slides over until its tail is on the tip of the other vector. The resultant is drawn from the
tail of the unmoved vector to the tip of the moved vector. Let’s subtract vector 𝐁 from
vector 𝐀, which are both shown on the grid.
Before we subtract our vectors,
let’s take a look at both their magnitudes and directions. Vector 𝐀 has a length of seven
units along the horizontal axis to the right of our screen, and vector 𝐁 has a
magnitude of three units along the horizontal axis to the right of our screen and
four units vertically to the top of our screen. To Find the vector that is formed
when we subtract vector 𝐁 from vector 𝐀, we start by drawing a vector that
represents the negative of vector 𝐁. To do this, we rotate vector 𝐁 180
degrees until it’s pointing in the opposite direction both horizontally and
vertically from where it was originally drawn. The length of the vector will still
be the same, even though the orientation has changed.
For our example, vector 𝐁 was
three units to the right and four units up. Therefore, vector negative 𝐁 would
be three units to the left and four units down. Another way to draw the negative
vector is to simply flip the tip and the tail of the original vector. In this case, vector negative 𝐁
would have its tail on the tip of vector 𝐁 and the tip of vector negative 𝐁 would
be on the tail of vector 𝐁. It doesn’t matter which method you
use as they both result in the negative of vector 𝐁.
For this example, we’ll use vector
negative 𝐁 that’s shown by the dotted yellow line. We can now apply the tip-to-tail
method. We leave vector 𝐀 where it is and
slide vector negative 𝐁 over until the tail of vector negative 𝐁 is on the tip of
vector 𝐀. It is important to slide over
vector negative 𝐁 and not vector 𝐁 because otherwise we would be adding our
original vectors together rather than subtracting them. Our resultant is drawn from the
tail of vector 𝐀 to the tip of vector negative 𝐁. The direction our resultant points
in is away from the origin or towards the tip of vector negative 𝐁. We can label the resultant vector
𝐕 as that is what we called our vector at the top of the screen. Our resultant vector has a length
of four units horizontally to the right and four units vertically to the bottom of
the screen.
Another way to subtract vectors is
to use unit vector notation. Let’s recall that a unit vector is
a vector of length one. Let’s take a look back at the three
vectors that we drew at the beginning of our video when we were refreshing our
memory on what a vector was and how to draw it graphically. Let’s start with vector 𝐀. We said that vector 𝐀 had a
magnitude of five units along the horizontal axis to the right of our screen. In unit vector notation, we would
say that vector 𝐀 has a value of five 𝐢 where the vector 𝐢 points in the
horizontal direction. The expression would be vector 𝐀
is equal to five 𝐢, where the 𝐢 has a little hat to represent that it’s a unit
vector. If this was in text form, our 𝐢
could be bolded to show that it’s a unit vector.
Vector 𝐁 also had a magnitude of
five units, except this time, it was along the vertical axis. In unit vector notation, we would
say that vector 𝐁 has a value of five 𝐣, where the vector 𝐣 points in the
vertical direction. The expression is vector 𝐁 is
equal to five 𝐣, with the 𝐣 having a little hat over it to represent its unit
vector notation. Vector 𝐂 had a magnitude of seven
units to the right along the horizontal axis and four units to the top of the screen
along the vertical axis. In unit vector notation, vector 𝐂
would have a value of seven 𝐢 plus four 𝐣, where the vector 𝐢 points in the
horizontal direction and the vector 𝐣 points in the vertical direction.
Now that we’ve done a quick recap
on unit vectors, let’s go over how to subtract vectors when they’re in unit vector
notation. When we’re subtracting vectors that
are in unit vector notation, we need to subtract the individual components. This means we have to subtract the
𝐢 hats from each other and the 𝐣 hats from each other separately. The resultant vector 𝐕 would have
an 𝐢-component that would be equal to the subtraction of the individual
𝐢-components and the 𝐣-component that would be equal to the subtraction of the
individual 𝐣-components.
If we have two vectors in unit
vector notation with vector 𝐀 being equal to eight 𝐢 plus 10 𝐣 and vector 𝐁
being equal to three 𝐢 plus two 𝐣, what would be the value of the resultant vector
that’s equal to vector 𝐀 minus vector 𝐁? Vector 𝐕 will be the resultant
vector of vector 𝐀 minus vector 𝐁. We will start by subtracting the
𝐢-components. Eight 𝐢 minus three 𝐢 is five
𝐢. Then, we subtract the
𝐣-components. 10𝐣 minus two 𝐣 is eight 𝐣. Our resultant vector 𝐕 has a value
of five 𝐢 plus eight 𝐣.
Now, let’s look what happens when
we switch the order of subtraction, meaning that this time, our resultant vector 𝐕
will be equal to vector 𝐁 minus vector 𝐀. Let’s look at it both graphically
and in unit vector notation. We’ll use the same vectors that we
did when we were subtracting graphically. We are subtracting vector 𝐀 from
vector 𝐁. So let’s start by writing vector 𝐁
in unit vector notation. Vector 𝐁 has a horizontal
component of three units to the right of the screen. Therefore, the 𝐢-component of
vector 𝐁 will be three. Vector 𝐁 also has a vertical
component of four units to the top of the screen. So, the 𝐣-component of vector 𝐁
will be four. Vector 𝐀 has a value of seven
units to the right horizontally. In unit vector notation, we would
say that vector 𝐀 is equal to seven 𝐢 plus zero 𝐣.
Let’s subtract our vectors
graphically. Remember that if we’re subtracting
vector 𝐀, that’s the same thing as adding negative vector 𝐀. Let’s remember from the beginning
of the video that drawing in the negative of a vector is the same thing as drawing
in a vector that’s 180 degrees to the original. Since vector 𝐀 has a value of
seven units to the right horizontally, vector negative 𝐀 will have a value of seven
units to the left horizontally. Then, we apply the tip-to-tail
method by sliding vector negative 𝐀 over until the tail of vector negative 𝐀 is on
the tip of vector 𝐁. Our resultant vector is drawn from
the tail of vector 𝐁 to the tip of vector negative 𝐀 and points away from the
origin or towards the tip of vector negative 𝐀.
We can see that our resultant
vector 𝐕 has a length of four units along the horizontal to the left of our screen
and a length of four units along the vertical to the top of the screen. In unit vector notation, our
resultant vector 𝐕 would be equal to negative four 𝐢 for the four units to the
left of our screen along the horizontal plus four 𝐣 for the four units along the
vertical to the top of our screen. We can verify our resultant vector
by subtracting vector 𝐀 from vector 𝐁, using unit vector notation.
Three 𝐢 minus seven 𝐢 is equal to
negative four 𝐢 and four 𝐣 minus zero 𝐣 is equal to four 𝐣. This matches the results that we’ve
got for the graphical method. Now let’s compare the resultant of
vector 𝐁 minus vector 𝐀 to our earlier resultant of vector 𝐀 minus vector 𝐁. We have drawn a pink dotted vector
to represent the resultant vector of 𝐀 minus 𝐁. Notice that the resultant vector of
𝐀 minus 𝐁 is the negative of the vector that represents 𝐁 minus 𝐀. If we were to write this as an
expression, we could say that vector 𝐀 minus vector 𝐁 is equal to the negative of
vector 𝐁 minus vector 𝐀.
Comparing the resultant vectors for
vector 𝐀 minus vector 𝐁, which is equal to four 𝐢 minus four 𝐣, and vector 𝐁
minus vector 𝐀, which is equal to negative four 𝐢 plus four 𝐣. We see once again that vector 𝐀
minus vector 𝐁 is equal to the negative of vector 𝐁 minus vector 𝐀. Now, let’s take a look at two
examples of subtracting vectors, one where we subtract the vectors graphically and
one where we subtract them using unit vector notation.
The diagram shows seven vectors 𝐀,
𝐁, 𝐏, 𝐐, 𝐑, 𝐒, and 𝐓. Which of the vectors is equal to 𝐀
minus 𝐁?
We are being asked to find the
vector resultant of vector 𝐀 minus vector 𝐁. Since our vectors have been given
to us graphically, we can solve for the vector subtraction using a graphical
method. We need to remember that when we’re
subtracting vector 𝐁 from vector 𝐀, it’s the same thing as adding the negative of
vector 𝐁 to vector 𝐀. To solve our problem, we are going
to need to draw in a vector that’s the negative of vector 𝐁. A negative vector is a vector that
is rotated 180 degrees from the original vector. Vector 𝐁 had a magnitude of three
units to the right of the screen. Therefore, vector negative 𝐁 will
have a value of three units to the left of the screen. Vector 𝐁 also had a length of five
units to the top of our screen and therefore vector negative 𝐁 will have a value of
five units to the bottom of the screen.
Now that we have drawn in vector
negative 𝐁, we can add vector 𝐀 and vector negative 𝐁 together using the
tip-to-tail method. In the tip-to-tail method, one
vector slides over until its tail is on the tip of the other vector. The resultant is drawn from the
tail of the unmoved vector to the tip of the moved vector. For our problem, we’re gonna slide
vector negative 𝐁 over until it’s on the tip of vector 𝐀. Then, we draw in a resultant vector
from the tail of vector 𝐀 to the tip of vector negative 𝐁. The resultant points away from the
origin towards the tip of vector negative 𝐁. We can see that vector 𝐐 overlaps
our resultant for vector 𝐀 minus vector 𝐁. Therefore, we can say that of the
seven vectors shown vector 𝐐 is equal to vector 𝐀 minus vector 𝐁.
Now, let’s subtract two vectors
using unit vector notation.
Consider the two vectors 𝐀 and 𝐁,
where 𝐀 is equal to eight 𝐢 plus 10𝐣 and 𝐁 is equal to three 𝐢 plus two 𝐣. Calculate 𝐀 minus 𝐁.
In our problem, we are given vector
𝐀 and 𝐁 in unit vector notation. Therefore, we need to remember that
when we subtract vectors in unit vector notation, we need to subtract the individual
components. This means that we’ll subtract the
𝐢’s and separately we’ll subtract the 𝐣’s. We can align the vectors vertically
along with the subtraction sign to make the subtraction of the components
easier. We put hats over our 𝐢 and 𝐣 unit
vectors as we are writing our workout by hand. But in the text form of the
problem, 𝐢 and 𝐣 were bolded to show that they were unit vectors.
We can use the vector 𝐕 to
represent the resultant vector of 𝐀 minus 𝐁. To find our resultant, we start
with the 𝐢-components. Eight 𝐢 minus three 𝐢 is five
𝐢. Then, we subtract the 𝐣’s. 10𝐣 minus two 𝐣 is eight 𝐣. When we subtract two vectors 𝐀 and
𝐁, where 𝐀 is equal to eight 𝐢 plus 10𝐣 and 𝐁 is equal to three 𝐢 plus two 𝐣,
we get a resultant of five 𝐢 plus eight 𝐣.
Key Points from the Lesson
When subtracting vectors
graphically, draw the negative vector for the vector being subtracted and then apply
the tip-to-tail method. When subtracting vectors in unit
vector notation, subtract the individual 𝐢-components and the individual
𝐣-components separately. Vector 𝐀 minus vector 𝐁 is equal
to the negative of vector 𝐁 minus vector 𝐀.
