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## Unit Vector

Vectors are geometric entities that have magnitude and direction. Vectors have a starting point and a terminal point which represents the final position of the point. Various arithmetic operations can be applied to vectors such as addition, subtraction, and multiplication. A vector that has a magnitude of 1 is termed a unit vector. For example, vector v = (1, 3) is not a unit vector, because its magnitude is not equal to 1, i.e., |v| = √(1^{2}+3^{2}) ≠ 1.

Any vector can become a unit vector when we divide it by the magnitude of the same given vector. A unit vector is also sometimes referred to as a direction vector. Let us learn more about the unit vector, its formula along with a few solved examples.

## What is Unit Vector?

A **unit vector** is a vector that has the magnitude equal to 1. The unit vectors are denoted by the “cap” symbol ^. The length of unit vectors is 1. Unit vectors are generally used to denote the direction of a vector. A unit vector has the same direction as the given vector but has a magnitude of one unit; For a vector **A**; a unit vector is; ^A and ^A=(1/|A|)^A

**i**, **j**, and **k** are the unit vectors in the directions of the x-axis, y-axis, and z-axis respectively in a 3-dimensional plane. i.e.,

- |
**i**| = 1 - |
**j**| = 1 - |
**k**| = 1

### Magnitude of a Vector

The magnitude of a vector gives the numeric value for a given vector. A vector has both a direction and a magnitude. The magnitude of a vector formula summarises the individual measures of the vector along the x-axis, y-axis, and z-axis. The magnitude of a vector **A** is |**A**|. For a given vector with the direction along the x-axis, y-axis, and z-axis, the magnitude of the vector can be obtained by calculating the square root of the sum of the squares of its direction ratios. Let us understand it clearly from the below magnitude of a vector formula.

For a vector **A** = a**i** + b**j** + c**k** its magnitude is:

|**A**| = √(a^{2} + b^{2} + c^{2})

For example, if **A** = 1**i** + 2**j** + 3**k**, then |**A**| = √(1^{2}+2^{2}+2^{2}) = √9 = 3 units.

## Unit Vector Notation

Unit Vector is represented by the symbol ‘^’, which is called a cap or hat, such as ^a. It is given by ^a = **a**/|**a**| Where |**a**| is for norm or magnitude of vector **a**. It can be calculated using a unit vector formula or by using a calculator.

### Unit vector in three-dimension

The unit vectors of **i**, **j**, and **k** are usually the unit vectors along the x-axis, y-axis, z-axis respectively. Every vector existing in the three-dimensional space can be expressed as a linear combination of these unit vectors. The dot product of two unit vectors is always a scalar quantity. On the other hand, the cross-product of two given unit vectors gives a third vector perpendicular (orthogonal) to both of them.

### Unit Normal Vector

A ‘normal vector’ is a vector that is perpendicular to the surface at a defined point. It is also called “normal” to a surface containing the vector. The unit vector that is acquired after normalizing the normal vector is the unit normal vector, also known as the “unit normal.” For this, we divide a non-zero normal vector by its vector norm.

## Unit Vector Formula

As vectors have both magnitude (value) and direction, they are shown with an arrow. In particular, ^a

denotes a unit vector. If we want to find the unit vector of any vector, we divide it by the vector’s magnitude. Usually, the coordinates of x, y, z are used to represent any vector.

A vector can be represented in two ways:

1. **a** = (x, y, z) using the brackets.

2. **a** = x**i** + y**j** +z**k**

The formula for the magnitude of a vector is:

|**a**|= **√**(x^{2} + y^{2} + z^{2})

The **formula of unit vector** in the direction of a given vector is:

- Unit Vector = Vector/Vector’s magnitude

## How to Calculate the unit vector?

To find a unit vector with the same direction as a given vector, simply divide the vector by its magnitude. For example, consider a vector **v** = (3, 4) which has a magnitude of |**v**|. If we divide each component of vector **v** by |**v**| to get the unit vector ^v

which is in the same direction as v.

|**v**| = **√**(3^{2} + 4^{2}) = 5

Thus, ^v = **v** / |**v**| = (3, 4) / 5 = (3/5, 4/5).

**How to represent vector in a bracket format?**

If **a** = (x, y, z), then the unit vector in the direction of **a** in bracket format is,

^a

= **a**/|**a**| = (x,y,z)/**√**(x^{2} + y^{2} + z^{2}) = ( x/ **√**(x^{2} + y^{2} + z^{2}), y/**√**(x^{2} + y^{2} + z^{2}), z/**√**(x^{2} + y^{2} + z^{2}) )

**How to represent vector in a unit vector component format?**

If **a** = x**i** + y**j** + z**k **is a vector then the unit vector in the direction of **a** in component format is,

^a = **a**/|**a**| = (x**i **+ y**j **+ z**k**)/ **√**(x^{2} + y^{2} + z^{2}) = x/**√**(x^{2} + y^{2} + z^{2}) . **i **+ y/**√**(x^{2} + y^{2} + z^{2}) . **j **+ z/**√**(x^{2} + y^{2} + z^{2}) . **k**

Where x, y, z represent the value of the vector along the x- axis, y-axis, z-axis respectively and

^a is a unit vector, **a** is a vector, |**a**| is the magnitude of the vector, **i**, **j**, **k** are the directed unit vectors along the x, y, and z axes respectively.

## 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.

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 **i**. The unit vector along the y-axis is represented by **j**, and the unit vector along the z-axis is represented by **k**.

The vector v can hence be written as:

**v** = x**i** + y**j** + z**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.

## Properties of Vectors

The properties of vectors are helpful to gain a detailed understanding of vectors and also to perform numerous calculations involving vectors, A few important properties of vectors are listed here.

**A**.**B**=**B**.**A****A**×**B**≠**B**×**A****i**.**i**=**j**.**j**=**k**.**k**= 1**i**.**j**=**j**.**k**=**k**.**i**= 0**i**×**i**=**j**×**j**=**k**×**k**= 0**i**×**j**=**k**;**j**×**k**=**i**;**k**×**i**=**j****j**×**i**= –**k**;**k**×**j**= –**i**;**i**×**k**= –**j**- The dot product of two vectors is a scalar and lies in the plane of the two vectors.
- The cross product of two vectors is a vector, which is perpendicular to the plane containing these two vectors.

## Examples on Unit Vector

**Example 1:** Find the unit vector which is in the direction of 3i + 4j – 5k.

**Solution:** Given vector **A** = 3**i** + 4**j** – 5**k**

Its magnitude is,

|**A**| = √(3^{2} + 4^{2} + (-5)^{2})

= √(9 + 16 + 25)

= √50

= 5√2

The unit vector in the direction of the given vector is,

= **A**/|**A**|

= (3**i** + 4**j** – 5**k**) / (5√2)

**Answer:** Hence the unit vector is (3/5√2) **i** + (4/5√2) **j** – (5/5√2) **k**.

**Example 2:** Find the vector of magnitude 8 units and in the direction of the vector **i** – 7**j** + 2**k**.

**Solution:** Given vector **A** = **i** – 7**j** + 2**k**.

|**A**| = √(1^{2} + (-7)^{2} + (2)^{2})

= √(1 + 49 + 4)

= √54

= 3√6

The unit vector can be calculated using this below formula.

= **A**/|**A**|

= (**i** – 7**j** + 2**k**) / (3√6)

The vector of magnitude 8 units = 8 × (**i** – 7**j** + 2**k**) / (3√6)

**Answer:** Therefore the vector of magnitude 8 units = (4√6/9) · (**i** – 7**j** + 2**k**).

**Example 3:** Find the unit vector parallel to the resultant of the vectors **A** = 2**i** – 3**j** + 4**k **and **B = **–**i** + 5**j** – 2**k**.

**Solution:**

The resultant vector of the given two vectors is:

**A** + **B** = (2**i** – 3**j** + 4**k**) + (-**i** + 5**j** – 2**k**) = **i** + 2**j** + 2**k**.

Its magnitude is,

|**A** + **B**| = √(1^{2} + 2^{2} + 2^{2})

= √(9)

= 3

To find the unit vector parallel to the resultant of the given vectors, we divide the above resultant vector by its magnitude. Thus, the required unit vector is,

(**A** + **B**) / |**A** + **B**| = (**i** + 2**j** + 2**k**) / 3 = 1/3 **i** + 2/3 **j** + 2/3 **k**

**Answer:** 1/3 **i** + 2/3 **j** + 2/3 **k**.

## FAQs on Unit Vector

### What is the Definition of Unit Vector?

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 **i**, **j**, **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

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

is obtained by dividing the vector **A** with its magnitude |**A**|. The unit vector has the same direction coordinates as that of the given vector. = **A**/|**A**|.

### What is a Normal Unit Vector?

A unit normal vector to a two-dimensional curve is a vector with magnitude 1 that is perpendicular to the curve at some point. Typically you look for a function that gives you all possible unit normal vectors of a given curve, not just one vector.

### How Do You Find the Unit Vector Perpendicular to Two Vectors?

The cross-product of two non-parallel results in a vector that is a vector that is perpendicular to both of them. So, for the given two vectors **x** and **y**, we know that, **x** × **y** will be a vector that is perpendicular to both **x** and **y**. Further, to find the unit vector of this resultant vector, we divide it by its magnitude. i.e., (**x** × **y**) / |**x** × **y**|, this would give the unit vector that is perpendicular to the given two vectors.

### When are the Two Vectors said to be Parallel Vectors?

Two or more vectors are parallel if they are moving in the same direction. Also, the cross-product of parallel vectors is always zero.

## Unit Vector Symbol

Unit vectors are usually determined to form the base of a vector space. Every vector in the space can be expressed as a linear combination of unit vectors. The dot products of two unit vectors is a scalar quantity whereas the cross product of two arbitrary unit vectors results in third vector orthogonal to both of them.

**What is the unit normal vector?**

The normal vector is a vector which is perpendicular to the surface at a given point. It is also called “normal,” to a surface is a vector. When normals are estimated on closed surfaces, the normal pointing towards the interior of the surface and outward-pointing normal are usually discovered. The unit vector acquired by normalizing the normal vector is the unit normal vector, also known as the “unit normal.”Here, we divide a nonzero normal vector by its vector norm.

## Unit Vector Formula

As explained above vectors have both magnitude (Value) and a direction. They are shown with an arrow.

The above is a unit vector formula.

**How to find the unit 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 u_{v} which is in the same direction as v.

**How to represent Vector in a bracket format?**

### Unit Vector Example

Here is an example based on the unit vector. Observe and follow each step and solve problems based on it.

**Question 1:**

**Magnitude of Unit Vector**

In order to calculate the numeric value of a given vector, the magnitude of the vector formula is used. The magnitude of a vector

*A*⃗ is |A|. The magnitude of a vector can be identified by calculating the square roots of the sum of squares of its direction vectors. The magnitude of a vector formula is given by:

will be read as A cap.

For a unit vector u in the same direction as vector v, we divide the vector by its magnitude

**Representation of a Vector**

There are two ways in which a vector can be represented-

**Solved Examples Related to Unit Vector**

**Unit Tangent Vector**

**Solved Examples**

**Example 1:**

Suppose,

There is a vector a= (1,0).

To find the magnitude of the given vector, we use the formula,

|a|= √(x^{2}+y^{2}+z^{2}) ……(1)

Using the magnitude of unit vector formula (1),

|a|= √ (1^{2}+0) =√1=1

Therefore, a is a unit vector.

**Example 2:**

Suppose,

There is a vector b= (2, 3).

To find the magnitude of the given vector, we use the formula,

|b|= √(x^{2}+y^{2}+z^{2}) ……(1)

Using the magnitude of unit vector formula (1),

|b|= √ (2^{2}+3^{2})

= √13

Now, √13≠1.

Hence, from the above calculations, we can conclude that,

a is not a unit vector.

### Formula To Find Unit Vector

â = a/|a|…….(2)

= (x,y,z)/√(x^{2}+y^{2}+z^{2})

For example, a= (12,4,3)

Then, using (1),

|a|= √ ((12^{2}) +(4^{2}) +(3^{2}))

= √144+16+9 = √169=13

Substituting, |a| in vector unit formula (2),

â = (12,4,3)/13

= ((12/13), (4/13), (3/13))

**Sample Problems**

**Problem 1. Find the unit vector of 2i + 4j + 5k.**

**Solution:**

**Problem 2. Find the unit vector of 3i + 4j + 5k.**

**Solution:**

**Problem 3. Find the unit vector of i + 2j + 2k.**

**Solution:**

**Problem 4. Find the unit vector of the resultant vector of i + 3j +5k and -j – 3k.**

**Solution:**

**Problem 5. Find the unit vector of 4i + 4j.**

**Solution:**

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