In this video, we’re going to learn
how to add two or more vectors, both graphically and using unit vector notation. We’ll start with the graphical
method. But before we learn how to add
vectors graphically, let’s refresh our memory on what a vector is and how we can
represent a vector in a graphical form. A vector is a quantity that has
both magnitude and direction. So an easy way of representing a
vector graphically is to use an arrow.
For example, this arrow represents
a vector with a magnitude of four units and the direction along the horizontal axis
toward the right of the screen. We can give this vector a name. Let’s call it 𝐀. And 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
denote vectors. And when we see a vector printed
online or in a book, for example, we often see that it’s written in bold to denote
the fact that it’s a vector.
Let’s draw another example vector,
and we’ll call this vector 𝐁. We can see that vector 𝐁 has the
same magnitude of four units as vector 𝐀. But this time, it has a different
direction. It’s oriented vertically toward the
top of the screen. Once again, when we label our
vector 𝐁, we draw a half arrow over the top of the variable to denote that it’s a
Before we move on to adding our
vectors, let’s draw one more example. Vector 𝐂 has components both in
the horizontal and vertical directions. In the horizontal direction, it has
a length of five units. And in the vertical direction, it
has a length of three units. So we could say that vector 𝐂 has
a horizontal component of five and a vertical component of three. We can see that vector 𝐂 has a
larger horizontal component than vector 𝐀 and it has a smaller vertical component
than vector 𝐁.
Okay, so now we’ve done a quick
recap on what vectors are and how we can represent them graphically. Let’s look at how we can add them
together, starting with a graphical method. Whenever we add vectors together,
for example, 𝐀 and 𝐁, the result will always be a vector as well. Let’s call this result vector
𝐕. The approach that we’ll use to add
together vectors graphically is called the tip-to-tail method. In this method, one vector slides
over until its tail is on the tip of the other vector. The resultant vector is drawn from
the tail of the unmoved vector to the tip of the moved vector.
Let’s try using this method to add
together two new vectors. Once again, we’ll call these
vectors 𝐀 and 𝐁. Since these vectors are different
to the vectors 𝐀 and 𝐁 that we used in the previous example, let’s quickly look at
their vertical and horizontal components before we add them. Vector 𝐀 has a horizontal
component of three and a vertical component of zero. And vector 𝐁 has a horizontal
component of one and a vertical component of three.
So to find the vector that’s formed
when we add vectors 𝐀 and 𝐁 together, we start by leaving vector 𝐀 where it is
and then sliding vector 𝐁 over so that the tail of vector 𝐁 touches the tip of
vector 𝐀. And when doing this, it’s very
important that we keep the horizontal and vertical components of 𝐁 the same. Once we’ve done this, we can draw
in the resultant vector, that is, the vector which is equivalent to the sum of 𝐀
and 𝐁. This is drawn from the tail of
vector 𝐀 to the tip of the moved vector 𝐁. And we can label this new vector 𝐕
as this is what we called it in our expression at the top of the screen.
We can think of vectors as
representing a movement of a certain distance in a certain direction. We can see here that if we started
at the origin and moved along vector 𝐀, then moved along vector 𝐁, we would end up
here. So overall, our movement is the
same as if we traveled along vector 𝐕. This is one way to think about what
it means when we say that 𝐕 is the sum of 𝐀 and 𝐁.
The tip-to-tail method isn’t
limited to just adding two vectors together. So in this example, let’s introduce
a third vector 𝐂. We can see that this vector has a
horizontal component of two to the left. So we could say it has a horizontal
component of negative two. It also has a vertical component of
one in the downward direction. So we could say a vertical
component of negative one. Let’s use the tip-to-tail method to
add together the vectors 𝐀, 𝐁, and 𝐂. And we’ll call the resultant vector
Once again, we’ll keep the vector
𝐀 stationary and we’ll slide along the vector 𝐁 so that its tail touches the tip
of vector 𝐀. Next, we slide along vector 𝐂 so
that the tail of vector 𝐂 touches the tip of the moved vector 𝐁. Now, we can draw in our resultant
from the origin or the tail of vector 𝐀 to the tip of vector 𝐂. We can see that the resultant
vector 𝐖 has a horizontal component of two and a vertical component of two. This method can continue for as
many vectors as we want to add together. Let’s move on now to a different
method, adding together vectors using unit vector notation.
First, we need to recall that a
unit vector is a vector of length one. When we’re looking at vectors in a
two-dimensional space, that is, where we have a horizontal and a vertical axis,
there are two special unit vectors that we need to be aware of, 𝐢 and 𝐣. We can see that when we write the
symbols for 𝐢 and 𝐣, we can put a little hat over the top of the letter to signify
that it’s a unit vector. For this reason, these unit vectors
are sometimes called 𝐢 hat and 𝐣 hat. And in text form, we’ll sometimes
see these vectors represented by an 𝐢 or a 𝐣 in bold as well.
We can see that the unit vector 𝐢
or 𝐢 hat is the unit vector that points in the 𝑥-direction, whereas the unit
vector 𝐣 or 𝐣 hat points in the 𝑦-direction. These vectors are useful because
any two-dimensional vector can be expressed in terms of 𝐢 and 𝐣. Specifically, we can express the
horizontal component in terms of 𝐢 and the vertical component in terms of 𝐣. For example, this vector 𝐀 has a
horizontal or 𝑥-component of positive one and a vertical or 𝑦-component of
positive three. This means that the vector 𝐀 is
equal to 𝐢 plus three times 𝐣.
To take another example, this
vector 𝐁 has an 𝑥-component of negative two and a 𝑦-component of negative
one. This means we can say 𝐁 is equal
to negative two times 𝐢 minus 𝐣. So expressing two-dimensional
vectors in terms of the unit vectors 𝐢 and 𝐣 gives us a way of separately
describing their horizontal components in terms of 𝐢 and their vertical components
in terms of 𝐣. We can use this to help us add
vectors together. To add vectors together using unit
vector notation, we simply need to add the horizontal components and the vertical
components separately. This means that for all the vectors
we’re adding together, we add their 𝐢-components to find the 𝐢-component of the
resultant vector and we add their 𝐣-components to find the 𝐣-component of the
So let’s say we have two vectors 𝐀
and 𝐁, and they’re both expressed using unit vector notation. Let’s say that 𝐀 is equal to four
𝐢 plus nine 𝐣 and 𝐁 is equal to seven 𝐢 plus five 𝐣. Let’s say we want to add these two
vectors together to find the resultant vector 𝐕. Well, to do this, we start by
adding the 𝐢’s together. Four 𝐢 plus seven 𝐢 gives us
11𝐢. Next, we add the 𝐣’s together. Nine 𝐣 plus five 𝐣 is 14𝐣. So here our resultant vector 𝐕 is
equal to 11𝐢 plus 14𝐣; that is, it has a horizontal component of 11 and a vertical
component of 14.
Let’s try a different example. This time, 𝐀 is equal to three 𝐢
plus five 𝐣 and 𝐁 is equal to four 𝐢 minus six 𝐣. The big difference in this example
is that one of our vectors has a negative component. The 𝑦-component of vector 𝐁 is
negative six 𝐣. However, even if we’re dealing with
negative components, we still add together the 𝐢- and 𝐣-components separately. We just need to take into account
any negative signs.
Dealing with the 𝐢’s first, we
have three 𝐢 plus four 𝐢, which gives us seven 𝐢. And now looking at the 𝐣’s, we
have five 𝐣 plus negative six 𝐣. Five 𝐣 plus negative six 𝐣 is
equal to five 𝐣 minus six 𝐣, which leaves us with negative one 𝐣. And the simplest way we can write
this is negative 𝐣. So in total, in this case, the
resultant vector 𝐕 is equal to seven 𝐢 minus 𝐣.
Let’s now take a look at two
examples of vector addition, one where we add them graphically and one where we add
them using unit vector notation.
Which of the vectors 𝐏, 𝐐, 𝐑,
𝐒, or 𝐓 shown in the diagram is equal to 𝐀 plus 𝐁?
So here we have a diagram showing
us seven vectors, including the vectors 𝐀 and 𝐁. And we’re being asked, of the other
five vectors 𝐏, 𝐐, 𝐑, 𝐒, and 𝐓, which one will be equal to the sum of 𝐀 and
𝐁. Since the vectors have been given
to us in a graphical form — they’re represented by the blue arrows — we can add them
together graphically 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.
In our problem, we want to add
together the vectors 𝐀 and 𝐁. So let’s keep the vector 𝐀
stationary and slide the vector 𝐁 over so that its tail rests on the tip of vector
𝐀, making sure that both the horizontal and vertical components of vector 𝐁 as
we’ve drawn it now are the same as the original vector 𝐁. In this case, the horizontal
component is two units to the right and the vertical component is three units
upward. We can now draw on the resultant
vector, that is, the vector which is equivalent to 𝐀 plus 𝐁. This is drawn from the tail of the
unmoved vector 𝐀 to the tip of the moved vector 𝐁.
We can see that this resultant
vector has the same direction and magnitude as the vector 𝐐. In other words, it’s equal to
vector 𝐐. This means that vector 𝐐 is equal
to the sum of 𝐀 and 𝐁. Of the five vectors drawn in the
diagram, 𝐏, 𝐐, 𝐑, 𝐒, and 𝐓, it’s vector 𝐐 that’s equal to 𝐀 plus 𝐁.
Now let’s look at another example
where we add vectors using unit vector notation.
Consider two vectors 𝐀 and 𝐁. 𝐀 is equal to two 𝐢 plus three
𝐣, and 𝐁 is equal to seven 𝐢 plus five 𝐣. Calculate 𝐀 plus 𝐁.
In this problem, the vectors 𝐀 and
𝐁 have been written in bold to show that they’re vectors. Since we’re going to be solving
this by hand, we can signify that 𝐀 and 𝐁 are vectors by drawing half arrows over
the top. Similarly, the unit vectors 𝐢 and
𝐣 have been written in bold. When writing these by hand, we draw
a hat over the top to signify that they’re unit vectors. Whenever we’re adding vectors which
are expressed in unit vector notation, like 𝐀 and 𝐁, we need to remember to add
the 𝐢-components and the 𝐣-components separately.
It can be useful to write the
vectors one above the other with a plus sign so that the 𝐢-components and the
𝐣-components are vertically aligned. Adding the two vectors 𝐀 and 𝐁
together will give us a vector as a result. We can give this resultant vector a
name. Let’s call it 𝐕. To determine the components of 𝐕,
we start by adding together the 𝐢-components of 𝐀 and 𝐁. Two 𝐢 plus seven 𝐢 is nine
𝐢. Next, we add together the
𝐣-components, and three 𝐣 plus five 𝐣 is eight 𝐣. If vector 𝐀 is equal to two 𝐢
plus three 𝐣 and vector 𝐁 is equal to seven 𝐢 plus five 𝐣, then 𝐀 plus 𝐁 is
equal to nine 𝐢 plus eight 𝐣.
Okay, now we’ve looked at a couple
of examples, let’s summarize the key points from this lesson. We’ve seen that we can add vectors
together graphically using the tip-to-tail method. We’ve also seen how to add together
vectors using unit vector notation. When we do this, we need to add the
𝐢-components together and the 𝐣-components together separately.