In this explainer, we will learn how to subtract one vector from another in two dimensions, using both graphical and algebraic methods.

Consider the vectors and shown on the grid below.

Recall that we can work out the result of graphically by moving vector so that the βtailβ of vector is positioned at the βtipβ of vector . This is shown on the grid below, and the result of adding the two vectors is .

But what would the result of *subtracting*Β
from be? What would the result of be?

Recall that the **negative** of a vector is a vector that has the same
*magnitude* but the opposite *direction*. The diagram below shows a
vector, , and its negative,
.

With ordinary numbers, subtracting a number is the same as adding the negative of that number. So, for example, is the same as

The same is true of vectors: is the same as

So, we can subtract one vector from another by first finding the negative of the vector we want to subtract and then adding the two vectors.

Going back to the first diagram, letβs change to .

We can now simply add these two vectors together by placing the tail of on the tip of . The result is vector shown on the grid below.

This is one way of subtracting vectors graphically, but there is another way, which is generally faster. Consider again the two vectors and , shown on the grid below.

The result of is a vector that goes from the tip of the second vector (in this case ) to the tip of the first vector (in this case ). This is vector , shown on the grid below.

If we think of vectors and
as being position vectors of points
and , then the result of
can be
thought of as a vector that goes β**from** Β Β **to**
Β β, as shown below.

Notice that, because of this, the result of
is **not** equal to the result of
. The result of
is a vector that goes from
to , as shown below.

As we can see from the diagram, changing the order of the two vectors in the subtraction produces the negative result. We can write this algebraically as

### Example 1: Subtracting Vectors Graphically

The diagram shows seven vectors: , , , , , , and .

- Which of the vectors is equal to ?
- Which of the vectors is equal to ?

### Answer

**Part 1**

Recall that we can find the result of by drawing a vector that goes from the tip of the second vector, , to the tip of the first vector, . This is shown on the grid below.

Comparing this to the other vectors shown in the diagram in the question, we see that the result of corresponds to vector .

**Part 2**

Recall that when we reverse the order of the vectors in a vector subtraction, we get the negative result. Expressed algebraically,

So, we can use our answer from part 1 to quickly work out the answer to this part. If , then

Looking back to the diagram in the question, we can see that the vector that is the negative of vector is vector .

### Example 2: Subtracting Vectors Graphically

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

### Answer

Recall that we can find the result of by drawing a vector that goes from the tip of the second vector, , to the tip of the first vector, . This is shown on the grid below.

Comparing this to the other vectors shown in the diagram in the question, we see that the result of corresponds to vector .

So far, we have looked at how to subtract vectors graphically, but we can also do this algebraically.

Recall that any vector can be represented as a sum of multiples of unit vectors along the and axes. If the horizontal component of is , the vertical component of is , is a unit vector along the -axis, and is a unit vector along the -axis, then

Similarly, if we say that then

If we expand out the brackets, we get

We can then gather like terms to get

So, if we know the components of two vectors, we can use this formula to subtract one vector from the other, and get the components of the resulting vector.

### Example 3: Subtracting Two Vectors given Their Components

Consider the two vectors and , where and . Calculate .

### Answer

We can use the formula to find the result of , where and are the horizontal components of the two vectors and and are the vertical components of the two vectors.

Substituting in the values given in the question, we get

The result of is .

### Example 4: Subtracting Two Vectors given Their Components

Consider the two vectors and . and . Which of the five vectors shown in the diagram is equal to ?

### Answer

In this question, we have to work out the result of the vector subtraction algebraically and then identify the resulting vector in the diagram.

We can use the formula to find the result of , where and are the horizontal components of the two vectors and and are the vertical components of the two vectors.

Substituting in the values given in the question, we get

The result of is . Looking at the diagram, we can see that this corresponds to vector .

### Key Points

- For two vectors and
, we can work out the result of
graphically by first finding the
**negative**of and then simply adding it to . - Another way we can think about doing vector subtraction graphically is to draw a new vector that goes from the βtipβ of vector to the βtipβ of vector .
- If we are given and in component form, we can use the formula to find the result of .