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.

### Watch This

Vector Addition: head-to-tail method

### Guidance

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 \begin{align*}b\end{align*} 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 \begin{align*}\vec{a}\end{align*} or \begin{align*}\vec{b}\end{align*}, or it can be shorter. Below are some more examples of the triangle method.

#### Example A

#### Example B

#### Example C

### Guided Practice

1. Vectors \begin{align*}\vec{m}\end{align*} and \begin{align*}\vec{n}\end{align*} are perpendicular. Make a diagram of each addition, find the magnitude and direction (with respect to \begin{align*}\vec{m}\end{align*} and \begin{align*}\vec{n}\end{align*}) of their resultant if \begin{align*}|\vec{m}| = 29.8|\vec{n}| = 37.7 \end{align*}

2. Vectors \begin{align*}\vec{m}\end{align*} and \begin{align*}\vec{n}\end{align*} are perpendicular. Make a diagram of each addition, find the magnitude and direction (with respect to \begin{align*}\vec{m}\end{align*} and \begin{align*}\vec{n}\end{align*}) of their resultant if \begin{align*}|\vec{m}| = 2.8|\vec{n}| = 5.4\end{align*}

3. Vectors \begin{align*}\vec{m}\end{align*} and \begin{align*}\vec{n}\end{align*} are perpendicular. Make a diagram of each addition, find the magnitude and direction (with respect to \begin{align*}\vec{m}\end{align*} and \begin{align*}\vec{n}\end{align*}) of their resultant if \begin{align*}|\vec{m}| = 11.9|\vec{n}| = 9.4\end{align*}

**Solutions:**

1. For the problem, use the Pythagorean Theorem to find the magnitude and \begin{align*}\tan \theta = \frac{|\vec{n}|}{|\vec{m}|}\end{align*}

magnitude \begin{align*}= 48.1\end{align*}, direction \begin{align*}= 51.7^\circ\end{align*}

2. For the problem, use the Pythagorean Theorem to find the magnitude and \begin{align*}\tan \theta = \frac{|\vec{n}|}{|\vec{m}|}\end{align*}

magnitude \begin{align*}= 6.1\end{align*}, direction \begin{align*}= 62.6^\circ\end{align*}

3. For the problem, use the Pythagorean Theorem to find the magnitude and \begin{align*}\tan \theta = \frac{|\vec{n}|}{|\vec{m}|}\end{align*}

magnitude \begin{align*}= 15.2\end{align*}, direction \begin{align*}= 38.3^\circ\end{align*}

### Concept Problem Solution

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 \begin{align*}45^\circ\end{align*}

### Explore More

\begin{align*}\vec{a}\end{align*} is in standard position with terminal point (1, 5) and \begin{align*}\vec{b}\end{align*} 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?

\begin{align*}\vec{c}\end{align*} is in standard position with terminal point (4, 3) and \begin{align*}\vec{d}\end{align*} 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?

\begin{align*}\vec{e}\end{align*} is in standard position with terminal point (3, 2) and \begin{align*}\vec{f}\end{align*} 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?

\begin{align*}\vec{g}\end{align*} is in standard position with terminal point (5, 5) and \begin{align*}\vec{h}\end{align*} 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?

\begin{align*}\vec{i}\end{align*} is in standard position with terminal point (1, 5) and \begin{align*}\vec{j}\end{align*} 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 \begin{align*}\vec{k}\end{align*} and \begin{align*}\vec{l}\end{align*} are perpendicular. Make a diagram of each addition, find the magnitude and direction (with respect to \begin{align*}\vec{k}\end{align*} and \begin{align*}\vec{l}\end{align*}) of their resultant if \begin{align*}|\vec{k}| = 42\end{align*} and \begin{align*}|\vec{l}| = 30 \end{align*}.