In this explainer, we will learn how to add two or more vectors in two dimensions, using both graphical and algebraic methods.

Recall that a **vector** is a quantity that has both a magnitude and a direction.

The grid below shows two vectors, represented by arrows.

The magnitude of each vector is represented by the *length* of each arrow. The two arrows shown in the diagram have the *same length* of four grid squares, so the two vectors have the *same magnitude*. However, they point in *different directions*. The blue vector points along the -axis, while the red vector points along the -axis.

The grid below shows two different vectors.

Both the green vector and the orange vector point in the *same direction*, but they have *different lengths*. The orange vector has a length of 3 grid squares, while the green vector has a length of 6 grid squares.

In this explainer, the symbol for a vector will be written with a half-arrow on top of it, for example, . In other media, you may see vectors notated in different ways, for example, in a bold font like .

Now look at the two vectors shown on the grid below.

What would the result of adding vectors and together be? We can work this out just using the diagram. Imagine moving vector so that the βtailβ of the arrow (the end *without* the arrowhead) is at the same point on the grid as the βtipβ of the arrow (the end *with* the arrowhead) that represents vector . This is shown in the diagram below.

Notice that the *length* and *direction* of vector have not changed. It has simply been *translated* across the grid. The sum of the two vectors is now given by a vector, , that goes from the βtailβ of vector to the βtipβ of vector , as shown by the magenta arrow in the diagram below.

We could also have done this the other way around. We could have moved the tail of to the tip of vector , and we would have gotten the same result, as shown below.

When adding vectors using this method, it does not matter which order we add them in, as long as we place the vectors tip-to-tail, without changing the length or direction of each vector.

We can also use this method to add *more* than two vectors together. The diagram below shows three vectors on a grid.

We can find the sum of the three vectors, , by placing them tip-to-tail, as shown below.

The resultant vector, , always goes from the tail of the first vector to the tip of the last vector. This method can be used to add any number of vectors.

Letβs have a look at some examples.

### Example 1: Adding Two Vectors Graphically

Which of the vectors , , , , and shown in the diagram is equal to ?

### Answer

Letβs start by redrawing the diagram, highlighting vectors and and graying out the rest.

We can find the sum of and graphically by moving vector so that the βtailβ of the arrow is on the βtipβ of the arrow that represents vector . This is shown below.

The resultant vector is the one that goes from the tail of vector to the tip of vector , which is vector .

### Example 2: Adding Three Vectors Graphically

Which of the vectors , , , , and shown in the diagram is equal to ?

### Answer

Letβs start by redrawing the diagram, highlighting vectors , , and and graying out the rest.

We can find the sum of , , and graphically by moving vectors and so that the βtailsβ of each arrow are on the βtipsβ of each preceding arrow. This is shown below.

The resultant vector is the one that goes from the tail of vector to the tip of vector , which is vector .

Recall that we can also represent vectors algebraically. In the diagram below, vector can be written as , where and are **unit vectors**. A unit vector is a vector with a length of 1 that points along one of the axes. The unit vector points along the -axis, and the unit vector points along the -axis. The horizontal component of has a length of 2 grid squares, so its horizontal component can be described as , or βtwo times the unit vector along the -axis.β The vertical component of has a length of 3 grid squares, so its vertical component can be described as , or βthree times the unit vector along the -axis.β Thus, vector .

If we know what the horizontal and vertical components of two or more vectors are, we can find the sum of those vectors algebraically.

The diagram below shows two vectors.

We can see from the diagram that vector has a length of 4 grid squares in the -direction, and 1 grid square in the -direction. Vector has a length of 3 grid squares in the -direction, and 3 grid squares in the -direction. We can write this as

In order to find what is, we simply add each of the -components and each of the -components, which gives us

Note that if one of the components of a vector has a minus sign in front of it, we must *include* this sign when we add the - and -components of two or more vectors together. For example, if
we should think of this as being
So if we were to add the two vectors
for the -components, we would add and 3, and we would get

Letβs have a look at some more example questions.

### Example 3: Adding Two Vectors Given in Component Form

Consider two vectors and , where and . Calculate .

### Answer

In order to find , we must add the -components of each vector together and the -components of each vector together, so

We now have the sum of these two vectors, written in component form.

### Example 4: Adding Two Vectors Given in Component Form

Consider two vectors: and . and . Calculate .

### Answer

In order to find , we must add the -components of each vector together and the -components of each vector together. We must not forget to include the minus signs before the numbers in our calculation. Doing this, we get

We now have the sum of these two vectors, written in component form.

We can also relate adding two vectors graphically to adding two vectors algebraically, as is the case in the next example question.

### Example 5: Adding Two Vectors Shown on a Diagram and Giving the Result in Component Form

The diagram shows two vectors: and . The grid squares in the diagram have a side length of 1. What is in component form?

### Answer

There are two ways we can approach solving this problem.

The first way is that we could add the vectors graphically and then find the components of the result. The diagram below shows the two vectors added together, where vector is moved so that its tail is on the tip of vector . Vector is the result.

We can see from the diagram that vector has a horizontal component of and a vertical component of , so it can be written as . This is our answer.

The second way we could have worked this out is simply by finding the vector components of and and then adding the -components of each vector together and the -components of each vector together. From the original diagram, we can see that so

As you can see, we get the same result. Whether we add two vectors graphically or algebraically, we are performing the same operation on the vectors.

### Key Points

- We can add two or more vectors graphically by placing each of the vectors so that the βtailβ of each vector is located at the βtipβ of the preceding vector.
- We can add two or more vectors algebraically by adding the -components of each vector together and the -components of each vector together.
- Adding vectors graphically and algebraically are two different ways of performing the same operation on vectors.