# Vector Calculator

## Solve vectors step by step

This calculator will add and subtract two vectors, with steps shown. It will also find the dot product, cross product, scalar projection, vector projection, angle betwen vectors, magnitude, unit vector, basis, orthonormal basis (using the Gram-Schmidt process), orthogonal complement, and triple product (both scalar and vector) and will multiply the vector by a scalar.

$\langle$ $\rangle$
Comma-separated.
$\langle$ $\rangle$
Comma-separated.

If the calculator did not compute something or you have identified an error, or you have a suggestion/feedback, please write it in the comments below.

The Vector Calculator is a multipurpose online resource created to handle vector operations. This tool makes vector calculations simple, allows you to get the results easily, and helps understand vectors better.

## How to Use the Vector Calculator?

• ### Input

Begin by entering your vector(s) in the specified field(s). Next, select the operation you wish to perform from the available options (such as addition, subtraction, dot product, or cross product).

• ### Calculation

Once you've input the data and chosen the operation, click on the "Calculate" button.

• ### Result

The Vector Calculator will quickly compute and display the result of the selected operation.

## What Are Vectors?

Vectors are crucial elements in mathematics and physics due to their unique properties. Unlike scalars, which only have a magnitude, vectors are defined by both a magnitude and a direction. This makes them indispensable for representing physical quantities like force, velocity, or displacement, which inherently have a directional component.

A vector in a two-dimensional (2D) space is typically represented in the following form:

$$\mathbf{\vec{u}}=a\mathbf{\vec{i}}+b\mathbf{\vec{j}}=\langle a,b\rangle$$

In this expression, $\mathbf{\vec{u}}$ is the vector, $a$ and $b$ are scalars, and $\mathbf{\vec{i}}$ and $\mathbf{\vec{j}}$ are unit vectors along the x and y axes, respectively. The unit vectors $\mathbf{\vec{i}}$ and $\mathbf{\vec{j}}$ essentially give direction to the vector, while $a$ and $b$ provide the magnitude in those directions.

For example, consider the vector $\mathbf{\vec{u}}=3\mathbf{\vec{i}}+3\mathbf{\vec{j}}$. Here, $3$ and $2$ are the magnitudes along the x and y axes, respectively. The vector $\mathbf{\vec{u}}$ essentially says, "Move 3 units in the x direction and 2 units in the y direction."

Vectors can also exist in three-dimensional (3D) spaces, where they are expressed as

$$\mathbf{\vec{u}}=a\mathbf{\vec{i}}+b\mathbf{\vec{j}}+c\mathbf{\vec{k}}=\langle a,b,c\rangle$$

Here, $c$ is the magnitude along the z-axis, and $\mathbf{\vec{k}}$ is the unit vector along that axis.

Vectors allow us to perform various mathematical operations like addition, subtraction, and multiplication (dot and cross). These operations are important in physics, engineering, computer science, and other scientific fields.

## How Can We Determine the Direction and Magnitude of a Vector?

The direction and magnitude of a vector are key components that define it. Here's how you can determine both:

• Magnitude of a Vector

The magnitude (or length) of the vector $\mathbf{\vec{u}}=a\mathbf{\vec{i}}+b\mathbf{\vec{j}}$ in a 2D space is given by the formula $\mathbf{\left\lvert\vec{u}\right\rvert}=\sqrt{a^2+b^2}$, where $\sqrt{}$ refers to the square root function. In a 3D space, for the vector $\mathbf{\vec{u}}=a\mathbf{\vec{i}}+b\mathbf{\vec{j}}+c\mathbf{\vec{k}}$, the magnitude is $\mathbf{\left\lvert\vec{u}\right\rvert}=\sqrt{a^2+b^2+c^2}$. The magnitude of a vector is always a non-negative number.

• Direction of a Vector

The direction of a vector is expressed in terms of an angle measured from a reference axis (like the x-axis). For the 2D vector $\mathbf{\vec{u}}=a\mathbf{\vec{i}}+b\mathbf{\vec{j}}$, the angle $\theta$ that the vector makes with the positive x-axis can be calculated using the formula $\theta=\operatorname{atan2}(b,a)$, where $\operatorname{atan2}$ is the two-argument inverse tangent function. The result is typically expressed in degrees or radians.

## Why Choose Our Vector Calculator?

• ### Efficiency

The calculator provides quick results, enabling you to save precious time and effort that manual calculations would demand, especially when dealing with complex vectors.

• ### Ease of Use

The calculator is designed to be simple to use, with an intuitive interface and straightforward instructions, making it accessible to both beginners and advanced users.

• ### Accuracy

With our Vector Calculator, you can be sure of precise results. It eliminates the potential for errors that can creep into manual calculations.

• ### Versatility

From calculating vector magnitudes to performing operations like addition, subtraction, dot product, or cross product, our calculator handles a wide range of vector operations.

### FAQ

#### What are vectors in math?

In mathematics, vectors are quantities that have both a magnitude (size) and a direction. They are used to represent physical quantities like force or velocity. Vectors are often represented in the form $a\mathbf{\vec{i}}+b\mathbf{\vec{j}}$ in a 2D space or $a\mathbf{\vec{i}}+b\mathbf{\vec{j}}+c\mathbf{\vec{k}}$ in a 3D space, where $a$, $b$, and $c$ are scalars and $\mathbf{\vec{i}}$, $\mathbf{\vec{j}}$, and $\mathbf{\vec{k}}$ are unit vectors along the x, y, and z axes, respectively.

#### What are the types of vectors?

There are several types of vectors, including:

• Zero Vector: Also known as a null vector, it has a magnitude of zero and no specific direction.
• Unit Vector: A vector with a magnitude of $1$, typically used to represent directions.
• Position Vector: A vector representing the position of a point in space relative to an origin.
• Co-Initial Vectors: Vectors that have the same initial point.
• Collinear Vectors: Vectors that lie along the same line.
• Equal Vectors: Vectors that have the same magnitude and direction.

#### Can I use this calculator for vectors in 3D space?

Yes, our Vector Calculator is capable of handling vector calculations in both 2D and 3D spaces.

#### What operations can the Vector Calculator perform?

The Vector Calculator can perform many operations, including vector addition, subtraction, dot product, and cross product.