# 5.14: Vector Addition

**At Grade**Created by: CK-12

**Practice**Vector Addition

You and a friend are pulling a box across a floor. However, each of you is pulling at a different angle. A diagram of your efforts looks like this:

Each of these forces is a vector. Can you determine the net force you and your friend are applying to the box? To find the net result of the effort, you need to add the vectors for each of the forces. By the end of this Concept, you'll be able to accomplish this.

### Adding Vectors

The sum of two or more vectors is called the **resultant** of the vectors. There are two methods we can use to find the resultant: the parallelogram method and the triangle method.

*The Parallelogram Method:* Another method we could use is the parallelogram method. To use the parallelogram method, we draw the vectors so that their initial points meet. Then, we draw in lines to form a parallelogram. The resultant is the diagonal from the initial point to the opposite vertex of the parallelogram. *It is important to note that we cannot use the parallelogram method to find the sum of a vector and itself*.

To find the sum of the resultant vector, we would again use a ruler and a protractor to find the magnitude and direction.

If you look closely, you’ll notice that the parallelogram method is really a version of the triangle or tip-to-tail method. If you look at the top portion of the figure above, you can see that one side of our parallelogram is really vector translated.

*The Triangle Method:* To use the triangle method, we draw the vectors one after another and place the initial point of the second vector at the terminal point of the first vector. Then, we draw the resultant vector from the initial point of the first vector to the terminal point of the second vector. *This method is also referred to as the tip-to-tail method*.

To find the sum of the resultant vector we would use a ruler and a protractor to find the magnitude and direction.

The resultant vector can be much longer than either or , or it can be shorter. Below are some more examples of the triangle method.

#### Solve the following problems below using the triangle method.

### Vocabulary

**Parallelogram Method:** The ** parallelogram method** is a method of adding vectors by creating a parallelogram from the two vectors being added.

**Resultant:** A ** resultant** is a vector representing the sum of two or more vectors.

**Triangle Method:** The ** triangle method** is a method of adding vectors by connecting the tail of one vector to the head of another vector.

### Examples

#### Example 1

Earlier, you were asked to determine the net force you and your friend are applying to the box.

A triangle method diagram of the vectors being added looks like this:

As you can see, the resultant force has a magnitude of 100 Newtons at an angle of

#### Example 2

Vectors and are perpendicular. Make a diagram of each addition, find the magnitude and direction (with respect to and ) of their resultant if

For the problem, use the Pythagorean Theorem to find the magnitude and

magnitude , direction

#### Example 3

Vectors and are perpendicular. Make a diagram of each addition, find the magnitude and direction (with respect to and ) of their resultant if

For the problem, use the Pythagorean Theorem to find the magnitude and

magnitude , direction

#### Example 4

Vectors and are perpendicular. Make a diagram of each addition, find the magnitude and direction (with respect to and ) of their resultant if

For the problem, use the Pythagorean Theorem to find the magnitude and

magnitude , direction

### Review

is in standard position with terminal point (1, 5) and is in standard position with terminal point (4, 2).

- Find the coordinates of the terminal point of the resultant vector.
- What is the magnitude of the resultant vector?
- What is the direction of the resultant vector?

is in standard position with terminal point (4, 3) and is in standard position with terminal point (2, 2).

- Find the coordinates of the terminal point of the resultant vector.
- What is the magnitude of the resultant vector?
- What is the direction of the resultant vector?

is in standard position with terminal point (3, 2) and is in standard position with terminal point (-1, 2).

- Find the coordinates of the terminal point of the resultant vector.
- What is the magnitude of the resultant vector?
- What is the direction of the resultant vector?

is in standard position with terminal point (5, 5) and is in standard position with terminal point (4, 2).

- Find the coordinates of the terminal point of the resultant vector.
- What is the magnitude of the resultant vector?
- What is the direction of the resultant vector?

is in standard position with terminal point (1, 5) and is in standard position with terminal point (-3, 1).

- Find the coordinates of the terminal point of the resultant vector.
- What is the magnitude of the resultant vector?
- What is the direction of the resultant vector?
- Vectors and are perpendicular. Make a diagram of each addition, find the magnitude and direction (with respect to and ) of their resultant if and .

### Answers for Review Problems

To see the Review answers, open this PDF file and look for section 5.14.

### Image Attributions

Here you'll learn how to add two vectors with either the triangle method or the parallelogram method.

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