Resolving vectors

In a nutshell

Resolving vectors is the process of turning a vector into two perpendicular components, which allows vectors to be added together when they are at any angle to each other.

Definitions

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Equations

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Resolving a vector

The two components of any vector is given by the following two equations.

​$$F_x = F\cos\theta$$​​

​$$F_y = F\sin\theta$$​​

where $$F_x$$ and $$F_y$$ are the horizontal and vertical components of the vector respectively, $$F$$ is the magnitude of the vector and $$\theta$$ is the angle the vector makes with the horizontal.​

1. Hypotenuse - $$F$$​​ 2. Opposite - $$F_y$$​​ 3. Adjacent - $$F_x$$​​

This graphic shows the locations of $$F_x$$, $$F_y$$ and $$F$$ on a triangle. The trigonometric ratios relating the lengths and angles of a triangle are as follows.

​$$\sin\theta = \dfrac{opposite}{hypotenuse}$$​​

​$$\cos\theta = \dfrac{adjacent}{hypotenuse}$$​​

​$$\tan\theta = \dfrac{opposite}{adjacent}$$​​

Substituting in $$F_x$$, $$F_y$$ and $$F$$ for the adjacent, opposite and hypotenuse of a triangle then rearranging the equations will give the equations for the components of a vector.​

Example

A force of magnitude $$10N$$ acts at $$60\degree$$ to the horizontal. Calculate the horizontal and vertical components.

First, write down the equations.

$$F_x=F\cos\theta$$

$$F_y=F\sin\theta$$​​

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Next, substitute in the values.

$$F_x = 10 \times \cos60$$

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$$F_x = 5N$$​​

$$F_y=F\sin\theta$$​​

$$F_y = 8.66N$$​​

The horizontal component is $$\underline{5N}$$ and the vertical component is $$\underline{8.66N}$$.

Adding non-perpendicular vectors

In order to add vectors that are not perpendicular to each other, the vectors must first be resolved into their components. Then, the total magnitudes in the horizontal and vertical directions are calculated. Finally, the components can be combined using Pythagoras' theorem to find the final magnitude and direction of the resultant.

Example

Calculate the resultant of the following two forces.

First, write down the equations to resolve the forces.

$$F_x = F\cos\theta$$​​

$$F_y = F\sin\theta$$​​

Next, substitute in the values.

$$F_x = 10\times\cos45$$​​

$$F_x = 7.1N$$​​

$$F_y = 10\times\sin45$$​​

$$F_y = 7.1N$$​​

Next, sum the horizontal and vertical forces.

$$R_x = 6 + 7.07$$​​

$$R_x = 13.1N$$​​

$$R_y = 7.1N$$​​

Next, use Pythagoras' theorem to calculate the resultant force.

$$R = \sqrt{7.1^2 + 13.1^2}$$​​

$$R = 14.9N$$​​

Finally, use trigonometry to calculate the resultant force's direction.

$$\tan\theta = \dfrac{opp}{adj}$$​​

$$\tan\theta = \dfrac{7.07}{13.07}$$​​

$$\theta = 28.4\degree$$​​

The resultant force has a magnitude of $$\underline{14.9N}$$ and a direction of $$\underline{28.4\degree}$$.