# Unit Vector Calculator

## Calculate unit vectors step by step

The calculator will find the unit vector in the direction of the given vector, with steps shown.

$\langle$ $\rangle$
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.

Introducing our Unit Vector Calculator, a robust tool designed specifically to help you determine the unit vector or normalized vector associated with a given vector. Unit vectors play a crucial role in representing direction and streamlining vector algebra computations by maintaining a magnitude of 1.

## How to Use the Unit Vector Calculator?

• ### Input

Begin by entering the coordinates of your vector into the designated field.

• ### Calculation

Click on the "Calculate" button. The calculator will swiftly compute the magnitude of the vector and display it for you. The calculator will then proceed to calculate the unit vector by dividing each coordinate of the vector by its magnitude. This process normalizes the vector, ensuring it has a magnitude of 1.

• ### Result

Finally, the unit vector will be presented to you, providing the normalized representation of the original vector. This unit vector retains the direction of the original vector but possesses a magnitude of 1.

## What Is a Unit Vector?

A unit vector, often denoted as $\mathbf{\vec{u}}$, is a vector with a magnitude of $1$. It can be obtained by normalizing a given vector $\mathbf{\vec{v}}$. Mathematically, the formula for calculating a unit vector is:

$$\mathbf{\vec{u}}=\frac{\mathbf{\vec{v}}}{\mathbf{\left\lvert\vec{v}\right\rvert}},$$

where $\mathbf{\vec{v}}$ is the original vector and $\mathbf{\left\lvert\vec{v}\right\rvert}$ represents the magnitude of $\mathbf{\vec{v}}$.

For example, let's consider the vector $\mathbf{\vec{v}}=\langle3,4\rangle$. To find the unit vector \mathbf{\vec{u}} corresponding to $\mathbf{\vec{v}}$, we calculate the magnitude of $\mathbf{\vec{v}}$ as $\mathbf{\left\lvert\vec{v}\right\rvert}=\sqrt{3^2+4^2}=\sqrt{25}=5$.

Now, we divide each coordinate of $\mathbf{\vec{v}}$ by its magnitude:

$$\mathbf{\vec{u}}=\left\langle\frac{3}{5},\frac{4}{5}\right\rangle$$

Thus, the unit vector $\mathbf{\vec{u}}$ for $\mathbf{\vec{v}}$ is $\left\langle\frac{3}{5},\frac{4}{5}\right\rangle$, representing the same direction as $\mathbf{\vec{v}}$ but with a magnitude of $1$.

Unit vectors are invaluable in various applications, such as representing directions in physics, finding orthogonal vectors, calculating dot products, and simplifying vector computations.

## How to Use the Unit Vector Formula?

To use the unit vector formula and find the unit vector of a given vector $\mathbf{\vec{v}}$, you can follow these steps:

1. Start by identifying the coordinates of the vector $\mathbf{\vec{v}}$. For example, let's assume $\mathbf{\vec{v}}=\left\langle v_1,v_2,v_3\right\rangle$.
2. Calculate the magnitude of $\mathbf{\vec{v}}$ by using the formula:

$$\mathbf{\left\lvert\vec{v}\right\rvert}=\sqrt{v_1^2+v_2^2+v_3^2}$$

Here, $v_1^2$ denotes $v_1$ raised to the power of $2$, and so on.

3. Divide each coordinate of $\mathbf{\vec{v}}$ by the magnitude $\mathbf{\left\lvert\vec{v}\right\rvert}$ to obtain the corresponding coordinates of the unit vector $\mathbf{\vec{u}}$:

$$\mathbf{\vec{u}}=\left\langle\frac{v_1}{\mathbf{\left\lvert\vec{v}\right\rvert}},\frac{v_2}{\mathbf{\left\lvert\vec{v}\right\rvert}},\frac{v_3}{\mathbf{\left\lvert\vec{v}\right\rvert}}\right\rangle$$

Each coordinate of $\mathbf{\vec{u}}$ is obtained by dividing the corresponding coordinate of $\mathbf{\vec{v}}$ by $\mathbf{\left\lvert\vec{v}\right\rvert}$.

4. The resulting vector $\mathbf{\vec{u}}$ is the unit vector corresponding to the original vector $\mathbf{\vec{v}}$.

By following these steps and utilizing the unit vector formula, you can find the unit vector for a given vector $\mathbf{\vec{v}}$. This normalized vector will have a magnitude of $1$ and the same direction as the original vector.

## Why Choose Our Unit Vector Calculator?

• ### Accuracy and Reliability

Our calculator uses precise mathematical algorithms to ensure accurate results. You can trust the calculations and rely on the provided unit vector values.

• ### Efficiency and Time-Saving

Our calculator offers a quick and efficient solution for finding unit vectors. It eliminates the need for manual calculations, saving you valuable time and effort.

• ### User-Friendly Interface

Our calculator features a user-friendly interface, making it easy for users of all levels of mathematical proficiency to input a vector and obtain a unit vector effortlessly.

• ### Versatility

Our calculator handles vectors of various dimensions, allowing you to work with vectors in two-dimensional, three-dimensional, or higher-dimensional spaces.

• ### Educational Resource

Our calculator not only provides the unit vector results but also aids in enhancing your understanding of unit vectors. It presents the step-by-step calculation process, enabling users to grasp the underlying concepts.

### FAQ

#### What is the magnitude of a unit vector?

The magnitude of a unit vector is always $1$.

#### What is a unit vector notation?

A unit vector notation is a way to represent a unit vector. It typically involves placing a caret (^) or a hat (ˆ) symbol on top of the vector variable to indicate that it represents a unit vector. For example, if the original vector is $\mathbf{\vec{v}}$, the corresponding unit vector would be denoted as $\mathbf{\hat{\vec{v}}}$.

#### How does the Unit Vector Calculator work?

The Unit Vector Calculator determines the unit vector or normalized vector of a given vector. It calculates the magnitude of the vector and then divides each coordinate by the magnitude to obtain the corresponding coordinates of the unit vector. The resulting unit vector maintains the direction of the original vector but has a magnitude of $1$.

#### Does the Unit Vector Calculator provide step-by-step explanations?

Yes, it provides step-by-step explanations.