In this explainer, we will learn how to add and subtract vectors in 2D.

We know that vectors can be represented by line segments with specific length (magnitude) and direction. We will use these to help visualize vector addition and subtraction.

The scope of this explainer will only consider vectors in two dimensions; however, the methodology described can be extended to vectors in three or more dimensions.

Recall that a unit vector is a vector with magnitude equal to 1 and that the unit vectors in the - and -directions are denoted by and respectively.

Any two-dimensional vector can be written in the form . These can alternatively be represented in component form as or .

### Definition: Vector Addition

Vector addition is the operation of adding two or more vectors to find their sum.

Given two (or more) vectors in component form, we can find their sum by adding the corresponding components of the vectors.

For example, if and , then .

Vector addition is the operation of adding two or more vectors together into a vector sum. The sum of two or more vectors is called the resultant.

We will now look at a couple of examples where we need to add vectors in two dimensions.

### Example 1: Finding the Sum of Two Vectors

If and , find .

### Answer

We recall that in Cartesian coordinates, vector addition can be performed by adding the corresponding components of the vectors.

If and , then .

In this question and .

So,

Thus, .

### Example 2: Finding the Components of Two Vectors and Their Sum from 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

Any two-dimensional vector can be written in terms of its - and -components in the form , where is the number of units in the positive -direction and is the number of units in the positive -direction.

From the initial point to the terminal point of , we travel 2 units right and 1 unit up. This corresponds to 2 units in the -direction and 1 unit in the -direction.

So,

From the initial point to the terminal point of , we travel 3 units left and 4 units down. This corresponds to units in the -direction and units in the -direction.

So,

We know that the sum of two vectors is known as the resultant and that in Cartesian coordinates, vector addition can be performed by adding the corresponding components of the vectors.

If and , then .

Since then

We could also read this information directly from the vector diagram.

From the initial point of to the terminal point of vector , we travel 1 unit left and 3 units down. This corresponds to unit in the -direction and units in the -direction.

So,

Thus, , , and .

Vector subtraction is the process of finding a vector difference; it is the inverse operation to vector addition. This means that . When subtracting from , we find the resultant of and .

### Definition: Vector Subtraction

Vector subtraction is the operation of subtracting two vectors to find their difference.

Given two vectors in component form, we can find their difference by subtracting the corresponding components of the vectors.

For example, if and , then .

It is worth noting that the effect of negating is a reversal in its direction. For example, if we had a vector , this would be a vector of length 5 parallel to the -axis pointing from left to right. If we negate , we get . The magnitude of the vector is unchanged; it is still parallel to the -axis, but its direction has reversed; it now points from right to left.

We will now look at some further examples where we will add and subtract vectors in two dimensions.

### Example 3: Subtracting Vectors Expressed in Terms of Unit Vectors

Given the vectors and , calculate .

### Answer

We begin by recalling that in Cartesian coordinates, vector subtraction can be performed by subtracting the corresponding components of the vectors.

If and , then .

So,

Thus, .

### Example 4: Adding and Subtracting Vectors

Given that , , and , find .

### Answer

We begin by recalling that in Cartesian coordinates, vector addition and subtraction can be performed by adding or subtracting the corresponding components of the vectors.

So,

Thus, .

### Example 5: Finding a Missing Vector given Another Vector and the Sum of the Two Vectors

Given that , and , find .

### Answer

We begin by recalling that in Cartesian coordinates, vector addition and subtraction can be performed by adding or subtracting the corresponding components of the vectors.

If and , then .

Since and , then

So, .

### Example 6: Finding the Sum of Two Vectors given One of Them and the Difference between Them

Given that and , find .

### Answer

We begin by recalling that in Cartesian coordinates, vector addition and subtraction can be performed by adding or subtracting the corresponding components of the vectors.

If and , then .

Since, and , then

We now calculate .

If and , then .

Since and , then

So, .

### Example 7: Finding a Vector given Another Two Vectors and an Expression between the Three Vectors

Given that , , and , find .

### Answer

If , , and , then .

Since , , and , then

So, .

While it is outside the scope of this explainer, we can represent vector
addition and subtraction graphically using either the *parallelogram method* or the
*triangle method*.

We will finish this explainer by recapping some of the key points.

### Key Points

- In Cartesian coordinates, vector addition and subtraction can be performed by adding or subtracting the corresponding components of the vectors.
- If and , then .
- If and , then .