# Vector Addition Calculator

This **vector addition calculator** will help you perform **vector addition** and **subtraction** in two and three dimensions and calculate the **resultant vector**. In this article, we shall learn to calculate vector sum using the vector addition formula. We shall also extend this knowledge to calculate vector subtraction.

## What is a vector?

We cannot express specific quantities with just one number in maths and physics. In physics, a **vector** is a quantity with both a **magnitude** and a **direction**, like displacement and force. Mathematically, a vector is either a row or a column matrix.

We can indicate that a quantity `a`

is a vector either by adding an arrow over its head $(\overrightarrow{a})$ or by using bold font $(\mathbf{a})$.

## Vector addition formula in two and three dimensions

When we add two vectors, we add the corresponding components of each vector to form a new * resultant vector*. For two dimensional vectors $\mathbf{a}$ and $\mathbf{b}$, we calculate the resultant vector as follows:

where:

- $\mathbf{c}$ - The
**resultant vector**of the vector sum; - $a_x, a_y$ - The two
**components**of**vector**$\mathbf{a}$; and - $b_x, b_y$ - the two
**components**of**vector**$\mathbf{b}$.

Extending the same to three dimensions, the vector addition formula to calculate the vector sum would be:

For two-dimensional vectors, there is another method to calculate vector addition when the **vector magnitude** and **direction (angle)** are known.

We must first determine the components of the vectors from their magnitude and direction:

where:

- $m$ - The
**magnitude**of the vector; and - $\theta$ - The
**angle**the vector forms with the horizontal axis.

Using this in the vector addition formula for two-dimensional vectors:

where:

- $m_a$ - The
**magnitude**of the vector $\mathbf{a}$; - $\theta_a$ - The
**angle**the vector $\mathbf{a}$ forms with the horizontal axis; - $m_b$ - The
**magnitude**of the vector $\mathbf{b}$; and - $\theta_b$ - The
**angle**the vector $\mathbf{b}$ forms with the horizontal axis;

If you need help calculating the sine and cosine of an angle, use our trigonometric functions calculator.

## Calculating vector subtraction

We can calculate vector subtraction similar to vector addition using the simple technique:

Hence we can consider this a vector addition of vector $\mathbf{a}$ and $\mathbf{-b}$ and calculate the resultant vector.

## How to use this vector addition calculator

This vector addition calculator is a versatile tool that can handle both two- and three-dimensional vector addition and subtraction. Also, you can input the vector magnitude and direction instead of components in the case of two-dimensional vectors.

**Specify**whether the vectors are**two-dimensional**(`2D`

) or**three-dimensional**(`3D`

);**Choose**between**vector addition**or**subtraction**.- You can also enter
**scalar coefficients**for the vector addition or subtraction. Choose the`with multiples`

option if there are any scalar coefficients and enter the corresponding values. - To use the vector coordinates for the calculation, select the
`Cartesian coordinates`

option and enter the vector components. To use the vector magnitude and direction, select the`magnitude and direction`

option and enter the corresponding values. - The vector addition calculator will automatically calculate the
**resultant vector**and display the output. You can view the answer in the**component**form or**magnitude and direction**form.

This calculator works backward too! You can enter the resultant vector and any of the two vectors to find the unknown vector.

**a**+

**b**.

**a**:

**b**:

**a**+

**b**: