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URL: https://nakafa.com/en/subjects/physics/vector/analytical-addition-subtraction
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Learn how to add and subtract vectors using x and y components, then determine the resultant magnitude and direction.

---

## Calculating a Resultant with Components

The graphical method helps us see the resultant direction, but its accuracy depends on the drawing. The analytical method is more precise because it uses component calculations.

The method is to resolve every vector into the $$x$$ and $$y$$ axes, add the components along each axis, and then rebuild the resultant.

Visible text: The method is to resolve every vector into the and axes, add the components along each axis, and then rebuild the resultant.

Component: MathContainer
Children:

```math
R_x=\sum F_x
```

```math
R_y=\sum F_y
```

> In the analytical method, vectors are added axis by axis, not directly from their arrow lengths.

## Steps for Vector Addition

Suppose several forces $$\vec{F}_1$$, $$\vec{F}_2$$, and $$\vec{F}_3$$ act on an object. Each force is first split into horizontal and vertical components.

Visible text: Suppose several forces , , and act on an object. Each force is first split into horizontal and vertical components.

After all components are known, the resultant components are:

Component: MathContainer
Children:

```math
R_x=F_{1x}+F_{2x}+F_{3x}+\cdots
```

```math
R_y=F_{1y}+F_{2y}+F_{3y}+\cdots
```

The resultant magnitude is:

```math
R=\sqrt{R_x^2+R_y^2}
```

The direction of the resultant relative to the positive $$x$$ axis can be found from:

Visible text: The direction of the resultant relative to the positive axis can be found from:

```math
\tan\theta=\frac{R_y}{R_x}
```

If $$R_x$$ or $$R_y$$ is negative, identify the quadrant first so the direction is not misplaced.

Visible text: If or is negative, identify the quadrant first so the direction is not misplaced.

## Two-Stage Displacement

A person walks in two stages. The first stage is equivalent to $$80\text{ m}$$ east and $$60\text{ m}$$ north. The second stage is equivalent to $$120\text{ m}$$ east and $$90\text{ m}$$ north.

Visible text: A person walks in two stages. The first stage is equivalent to east and north. The second stage is equivalent to east and north.

| Displacement vector | $$x$$ component | $$y$$ component |
| :------------------ | :-------------------------------- | :-------------------------------- |
| $$\vec{s}_1$$ | $$80\text{ m}$$ | $$60\text{ m}$$ |
| $$\vec{s}_2$$ | $$120\text{ m}$$ | $$90\text{ m}$$ |
| $$\vec{R}$$ | $$200\text{ m}$$ | $$150\text{ m}$$ |

Visible text: | Displacement vector | component | component |
| :------------------ | :-------------------------------- | :-------------------------------- |
| | | |
| | | |
| | | |

The resultant components are:

Component: MathContainer
Children:

```math
R_x=80+120=200\text{ m}
```

```math
R_y=60+90=150\text{ m}
```

The total displacement magnitude is:

```math
\begin{aligned}
R&=\sqrt{200^2+150^2} \\
&=\sqrt{40000+22500} \\
&=250\text{ m}
\end{aligned}
```

The resultant direction is:

```math
\tan\theta=\frac{150}{200}=0.75
```

Therefore $$\theta\approx36.9^\circ$$ above east. The total displacement is $$250\text{ m}$$ at $$36.9^\circ$$ north of east.

Visible text: Therefore above east. The total displacement is at north of east.

## Subtraction with Components

Vector subtraction is also done component by component. If $$\vec{C}=\vec{A}-\vec{B}$$, then:

Visible text: Vector subtraction is also done component by component. If , then:

Component: MathContainer
Children:

```math
C_x=A_x-B_x
```

```math
C_y=A_y-B_y
```

For example, let $$\vec{A}=7\hat{i}+4\hat{j}$$ and $$\vec{B}=2\hat{i}+6\hat{j}$$.

Visible text: For example, let and .

```math
\begin{aligned}
\vec{C}&=(7-2)\hat{i}+(4-6)\hat{j} \\
&=5\hat{i}-2\hat{j}
\end{aligned}
```

The negative sign in $$-2\hat{j}$$ means the resulting $$y$$ component points downward.

Visible text: The negative sign in means the resulting component points downward.

So subtraction does not need a new rule. Keep the same axes, subtract matching components, then read the signs to recover the direction of the result.