# Unit Vector Formula

A unit vector formula is used to find the unit vector of the given vector. The given vector is divided by the magnitude of the vector, to obtain the unit vector. The unit vector has all the same vector components of the given vector but has a magnitude of one. The unit vector formula uses the concept of the magnitude of the vector.

## What is Unit Vector Formula?

The unit vector \(\hat{A}\) is obtained by dividing the vector \(\vec{A}\) with its magnitude |\(\vec{A}\)|. The unit vector has the same direction coordinates as that of the given vector.

### \(\hat{A}\) = \(\vec{A}\)/ |\(\vec{A}\)|

Let us try out a few examples to understand how to use the unit vector formula.

**Unit Vector Definition: **Vectors that have magnitude equals to 1 are called unit vectors, denoted by \(\hat{A}\). It is also called the multiplicative identity of vectors. The length of unit vectors is 1. It is generally used to denote the direction of a vector.

## Application of Unit Vector

Unit vectors specify the direction of a vector. Unit vectors can exist in both two and three-dimensional planes. Every vector can be represented with its unit vector in the form of its components. The unit vectors of a vector are directed along the axes. Unit vectors in 3-d space can be represented as follows: \(v = \hat{x} + \hat{y} + \hat{z}\)

In the 3-d plane, the vector **v **will be identified by three perpendicular axes (x, y, and z-axis). In mathematical notations, the unit vector along the x-axis is represented by \(\hat{i}\). The unit vector along the y-axis is represented by \(\hat{j}\), and the unit vector along the z-axis is represented by \(\hat{k}\).

The vector v can hence be written as:

v = \(x\hat{i}\) + \(y\hat{j}\) + \(z\hat{k}\)

Electromagnetics deals with electric forces and magnetic forces. Here vectors come in handy to represent and perform calculations involving these forces. In day-to-day life, vectors can represent the velocity of an airplane or a train, where both the speed and the direction of movement are needed.

## Examples Using Unit Vector Formula

**Example 1: Find the unit vector of \(3\hat{i} + 4\hat{j} - 5\hat{k}\).**

**Solution:** Given vector \(\vec{A}\) = \(3\hat{i} + 4\hat{j} - 5\hat{k}\)

|\(\vec{A}\)| = √{3^{2} + 4^{2} + (-5)^{2}} = √{9 + 16 + 25} = √{50} = 5√2

\(\hat{A}\) = (1/|\(\vec{A}\)|).\(\vec{A}\) = \(3\hat{i} + 4\hat{j} - 5\hat{k}\)

**Answer:** Hence the unit vector is (1/5√2). \(3\hat{i} + 4\hat{j} - 5\hat{k}\).

**Example 2: Find the vector of magnitude 8 units and in the direction of the vector \( \hat i - 7\hat j + 2\hat k\).**

**Solution:** Given vector \(\vec A = \hat i - 7\hat j + 2\hat k \).

\(\begin{align}|\vec{A}| &= \sqrt{1^2 + (-7)^2 + 2^2} \\&= \sqrt{1 + 49 + 4} \\&= \sqrt{54}\\&=3\sqrt6\end{align}\)

The unit vector can be calculated using this below formula.

\(\begin{align}\hat A &= \frac{1}{|\vec{A}|}.\vec A \\&= \frac{1}{3\sqrt6}.(\hat i - 7\hat j + 2\hat k)\end{align}\)

The vector of magnitude 8 units = \(\frac{4\sqrt6}{9}.(\hat i - 7\hat j + 2\hat k)\)

**Answer:** Therefore the vector of magnitude 8 units = \(\frac{4\sqrt6}{9}.(\hat i - 7\hat j + 2\hat k)\)

**Example 3: How to find a unit vector u that has the same direction as vector a = 10i + 24j?**

**Solution:** To calculate the unit vector of a = 10i + 24j, we follow the below steps:

Find the magnitude of the given vector |a| = √(10^{2} + 24^{2}) = √676 = 26

Now, we have the formula: a / |a| = u

Using the above formula, we get:

u = (10i + 24j) / 26

u = (5/13)i + (12/13)j

**Answer:** Hence, the unit vector u that has the same direction as vector a = 10i + 24j is u = (5/13)i + (12/13)j.

## FAQs on Unit Vector Formula

### What Is a Unit Vector in Math?

A vector that has a magnitude of 1 is a unit vector. It is also known as a direction vector because it is generally used to denote the direction of a vector. The vectors \(\hat{i}\), \(\hat{j}\), \(\hat{k}\), are the unit vectors along the x-axis, y-axis, and z-axis respectively.

How Do You Find the Unit Vector With the Same Direction as a Given Vector?

To find a unit vector with the same direction as a given vector, we divide the vector by its magnitude. For example, consider a vector v = (1, 4) which has a magnitude of |v|. If we divide each component of vector v by |v| we will get the unit vector \(\hat{v}\) which is in the same direction as v.

What Is a Unit Vector Used For?

Unit vectors are only used to specify the direction of a vector. Unit vectors exist in both two and three-dimensional planes. Every vector has a unit vector in the form of its components. The unit vectors of a vector are directed along the axes.

What Is a Unit Vector Formula?

The unit vector \(\hat{A}\) is obtained by dividing the vector \(\vec{A}\) with its magnitude |A|. The unit vector has the same direction coordinates as that of the given vector. \(\hat{A}\) = A/|A|