Created by Krishna Nelaturu
Last updated: Jul 10, 2022

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:

\begin{align*} \mathbf{c} &= \mathbf{a} + \mathbf{b}\\ &= \begin{bmatrix} a_x \\ a_y \end{bmatrix} + \begin{bmatrix} b_x \\ b_y \end{bmatrix}\\ \mathbf{c} &= \begin{bmatrix} a_x + b_x\\ a_y + b_y \end{bmatrix} \end{align*}

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:

\begin{align*} \mathbf{c} &= \mathbf{a} + \mathbf{b}\\ &= \begin{bmatrix} a_x \\ a_y\\ a_z \end{bmatrix} + \begin{bmatrix} b_x \\ b_y\\ b_z \end{bmatrix}\\ \mathbf{c} &= \begin{bmatrix} a_x + b_x\\ a_y + b_y\\ a_z + b_z \end{bmatrix} \end{align*}

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:

$a_x = m\cdot \cos\theta\\ a_y = m\cdot \sin\theta$

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:

\begin{align*} \mathbf{c} &= \begin{bmatrix} a_x + b_x\\ a_y + b_y \end{bmatrix} \\ \mathbf{c}&= \begin{bmatrix} m_a\cdot \cos\theta_a + m_b\cdot \cos\theta_b\\ m_a\cdot \sin\theta_a + m_b\cdot \sin\theta_b \end{bmatrix} \end{align*}

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:

$\mathbf{c} = \mathbf{a} - \mathbf{b} = \mathbf{a} + \mathbf{(-b)}$

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.

Krishna Nelaturu
Vectors in
2D
Operation
without
multiples
Perform vector addition a + b.
Vector a:
Cartesian coordinates
x
y
Vector b:
Cartesian coordinates
x
y
Result
Vector a + b:
Cartesian coordinates
x
y
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