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URL: https://nakafa.com/en/subjects/mathematics/vector-operations/unit-vector
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/material/lesson/mathematics/vector-operations/unit-vector/en.mdx

Calculate unit vectors from any vector through normalization. Learn direction representation with magnitude 1, standard unit vectors î, ĵ, k̂.

---

## Definition of Unit Vector

A unit vector is a vector that has a length or magnitude equal to $$1 \text{ unit}$$. Unit vectors are used to indicate the direction of a vector in space. If we have a vector, we can obtain a unit vector with the same direction by dividing the vector by its length.

Visible text: A unit vector is a vector that has a length or magnitude equal to . Unit vectors are used to indicate the direction of a vector in space. If we have a vector, we can obtain a unit vector with the same direction by dividing the vector by its length.

## Basic Concept

Consider vector $$\overrightarrow{PQ}$$ in a Cartesian coordinate system. The unit vector of $$\overrightarrow{PQ}$$ is defined as vector $$\overrightarrow{PQ}$$ divided by its length.

Visible text: Consider vector in a Cartesian coordinate system. The unit vector of is defined as vector divided by its length.

```math
\vec{a}_{PQ} = \frac{\overrightarrow{PQ}}{|\overrightarrow{PQ}|}
```

Where $$|\overrightarrow{PQ}|$$ is the length or magnitude of vector $$\overrightarrow{PQ}$$.

Visible text: Where is the length or magnitude of vector .

A unit vector always points in the same direction as the original vector but has its length normalized to $$1 \text{ unit}$$.

Visible text: A unit vector always points in the same direction as the original vector but has its length normalized to .

Component: Vector3d
Props:
- title: Vector and Unit Vector
- description: Visualization of the original vector and the unit vector that have the same direction but different lengths.
- vectors: [
{
from: [0, 0, 0],
to: [3, 4, 0],
color: getColor("PURPLE"),
label: "v",
labelPosition: "end",
},
{
from: [0, 0, 0],
to: [0.6, 0.8, 0],
color: getColor("AMBER"),
label: "v̂",
labelPosition: "end",
},
]
- cameraPosition: [6, 6, 8]

## Calculating a Unit Vector

### Calculation Steps

To determine the unit vector of a given vector, follow these steps:

1. Identify the original vector
2. Calculate the length of the vector
3. Divide the vector by its length

### Example Application

Let's say we have vector $$\vec{v} = (3, 6, 4)$$

Visible text: Let's say we have vector

First step, we calculate the length of the vector:

```math
|\vec{v}| = \sqrt{3^2 + 6^2 + 4^2} = \sqrt{9 + 36 + 16} = \sqrt{61}
```

Second step, we divide the vector by its length:

```math
\hat{v} = \frac{\vec{v}}{|\vec{v}|} = \frac{(3, 6, 4)}{\sqrt{61}}
```

Therefore, we get:

```math
\hat{v} = \left(\frac{3}{\sqrt{61}}, \frac{6}{\sqrt{61}}, \frac{4}{\sqrt{61}}\right)
```

Component: ContentBlock
Children:
Component: Vector3d
Props:
- title: Example of Unit Vector in 3D Space
- description: Visualization of vector $$v(3,6,4)$$ and its unit
vector $$v̂$$ pointing in the same direction.
  Visible text: Visualization of vector and its unit
vector pointing in the same direction.
- vectors: [
{
from: [0, 0, 0],
to: [3, 6, 4],
color: getColor("PINK"),
label: "v",
labelPosition: "end",
},
{
from: [0, 0, 0],
to: [3 / Math.sqrt(61), 6 / Math.sqrt(61), 4 / Math.sqrt(61)],
color: getColor("TEAL"),
label: "v̂",
labelPosition: "end",
},
]
- cameraPosition: [4, 8, 8]

## Properties of Unit Vectors

### Length of a Unit Vector

A unit vector always has a length equal to $$1$$. This can be proven by calculating the length of the unit vector:

Visible text: A unit vector always has a length equal to . This can be proven by calculating the length of the unit vector:

```math
|\hat{v}| = \sqrt{\left(\frac{3}{\sqrt{61}}\right)^2 + \left(\frac{6}{\sqrt{61}}\right)^2 + \left(\frac{4}{\sqrt{61}}\right)^2} = \sqrt{\frac{9 + 36 + 16}{61}} = \sqrt{\frac{61}{61}} = 1
```

### Unit Vectors on Coordinate Axes

In a three-dimensional coordinate system, there are three standard unit vectors that are parallel to each coordinate axis:

- $$\hat{\imath}$$ is the unit vector in the direction of the $$x$$-axis
- $$\hat{\jmath}$$ is the unit vector in the direction of the $$y$$-axis
- $$\hat{k}$$ is the unit vector in the direction of the $$z$$-axis

Visible text: - is the unit vector in the direction of the -axis
- is the unit vector in the direction of the -axis
- is the unit vector in the direction of the -axis

Any vector can be expressed as a linear combination of these three unit vectors.

Component: Vector3d
Props:
- title: Standard Unit Vectors
- description: Standard unit vectors on the coordinate axes.
- vectors: [
{
from: [0, 0, 0],
to: [1, 0, 0],
color: getColor("ROSE"),
label: "î",
labelPosition: "end",
},
{
from: [0, 0, 0],
to: [0, 1, 0],
color: getColor("VIOLET"),
label: "ĵ",
labelPosition: "end",
},
{
from: [0, 0, 0],
to: [0, 0, 1],
color: getColor("ORANGE"),
label: "k̂",
labelPosition: "end",
},
]
- cameraPosition: [3, 3, 3]

## Applications of Unit Vectors

### Indicating Direction

Unit vectors are very useful for indicating direction without regard to magnitude or length. In physics, for example, unit vectors are often used to indicate the direction of force, velocity, or acceleration.

### Physics Calculations

In physics, when we want to decompose a vector into its components, unit vectors are very helpful. For instance, a force can be decomposed into components along the x, y, and z axes using unit vectors.