# 5.20: Resultant as the Sum of Two Components

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

**Practice**Resultant as the Sum of Two Components

You are working in science class on a "weather unit". As part of this class, you are tasked with going out and checking the wind speed each day at a meter behind your school. The wind speed you record for the day is 20 mph at a

### Resultant of the Sum of Two Components

We can look at any vector as the resultant of two perpendicular components. If we generalize some vector

If we are given the vector

This is accomplished by taking the magnitude of the vector times the cosine of the vector's angle to find the horizontal component, and the magnitude of the vector times the sine of the vector's angle to find the vertical component.

#### Finding the Horizontal and Vertical Components

1. If

If we know an angle and a side of a right triangle, we can find the other remaining sides using trigonometric ratios. In this case,

To find

The horizontal component is 5.7 and the vertical component is 18.7. One can rewrite this in vector notation as

2. If

To find

#### Finding the Resultants

If

We can view each of these vectors on the coordinate system here:

Each of these vectors then serves as sides in a right triangle. So we can use the Pythagorean Theorem to find the length of the resultant:

The angle of rotation that the vector makes with the "x" axis can be found using the tangent function:

### Examples

#### Example 1

Earlier, you were asked to break down vectors into their individual components.

From this section you've learned how to take a vector and break it into components using trig functions. If you draw the wind speed you recorded as a vector:

You can find the "x" and "y" components. These are the same as the part of the wind that is blowing to the East and the part of the wind that is blowing to the North.

East component:

x = 12.86 mph

North component:

y = 15.32 mph

#### Example 2

Find the magnitude of the horizontal and vertical components of the following vector if the resultant vector’s magnitude and direction are given as

#### Example 3

Find the magnitude of the horizontal and vertical components of the following vector if the resultant vector’s magnitude and direction are given as \begin{align*}\text{magnitude} = 3.4 \qquad \qquad \quad \text{direction} = 162^\circ\end{align*}.

\begin{align*}\cos 162^\circ = \frac{x}{3.4}, \sin 162^\circ = \frac{y}{3.4}, x = 3.2, y = 1.1\end{align*}

#### Example 4

Find the magnitude of the horizontal and vertical components of the following vector if the resultant vector’s magnitude and direction are given as \begin{align*}\text{magnitude} = 15.9 \qquad \qquad \ \text{direction} = 12^\circ\end{align*}.

\begin{align*}\cos 12^\circ = \frac{x}{15.9}, \sin 12^\circ = \frac{y}{15.9}, x = 15.6, y = 3.3\end{align*}

### Review

Find the horizontal and vertical components of the following vectors given the resultant vector’s magnitude and direction.

- \begin{align*}\text{magnitude} = 65 \quad \text{direction} = 22^\circ\end{align*}.
- \begin{align*}\text{magnitude} = 34 \quad \text{direction} = 15^\circ\end{align*}.
- \begin{align*}\text{magnitude} = 29 \quad \text{direction} = 160^\circ\end{align*}.
- \begin{align*}\text{magnitude} = 100 \quad \text{direction} = 320^\circ\end{align*}.
- \begin{align*}\text{magnitude} = 320 \quad \text{direction} = 200^\circ\end{align*}.
- \begin{align*}\text{magnitude} = 15 \quad \text{direction} = 110^\circ\end{align*}.
- \begin{align*}\text{magnitude} = 10 \quad \text{direction} = 80^\circ\end{align*}.
- \begin{align*}\text{magnitude} = 90 \quad \text{direction} = 290^\circ\end{align*}.
- \begin{align*}\text{magnitude} = 87 \quad \text{direction} = 10^\circ\end{align*}.
- \begin{align*}\text{magnitude} = 42 \quad \text{direction} = 150^\circ\end{align*}.

- If \begin{align*}|\vec{r}| = 12\end{align*} and \begin{align*}|\vec{s}| = 8\end{align*}, find the resultant vector magnitude and angle.
- If \begin{align*}|\vec{r}| = 14\end{align*} and \begin{align*}|\vec{s}| = 6\end{align*}, find the resultant vector magnitude and angle.
- If \begin{align*}|\vec{r}| = 9\end{align*} and \begin{align*}|\vec{s}| = 24\end{align*}, find the resultant vector magnitude and angle.
- Will cosine always be used to find the horizontal component of a vector?
- If you know the component form of a vector, how can you find its magnitude and direction?

### Review (Answers)

To see the Review answers, open this PDF file and look for section 5.20.

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### Image Attributions

Here you'll learn how to express a vector as the sum of two component vectors.

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