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

Learn to break down vectors into x, y, z components, calculate magnitude using Pythagorean theorem, and find unit direction vectors with examples.

---

## Understanding Vector Components

In studying vectors, we need to understand that every vector can be broken down into its components. Vector components are parts of a vector that are parallel to the coordinate axes.

Vector components are values that indicate how far a vector moves in the direction of the $$x$$-axis and $$y$$-axis. Every vector in a plane can be expressed as a linear combination of unit vectors $$i$$ and $$j$$.

Visible text: Vector components are values that indicate how far a vector moves in the direction of the -axis and -axis. Every vector in a plane can be expressed as a linear combination of unit vectors and .

Component: Vector3d
Props:
- title: Vector Visualization and Its Components
- description: Vector $$AB$$ and its components on the{" "}
$$x$$, $$y$$, and{" "}
$$z$$ axes.
  Visible text: Vector and its components on the{" "}
, , and{" "}
 axes.
- vectors: [
{
from: [0, 0, 0],
to: [4, 6, 0],
color: getColor("ROSE"),
label: "AB",
},
{
from: [0, 0, 0],
to: [4, 0, 0],
color: getColor("TEAL"),
label: "x component",
},
{
from: [0, 0, 0],
to: [0, 6, 0],
color: getColor("LIME"),
label: "y component",
},
]

If we have a vector $$\overrightarrow{AB}$$, then:

Visible text: If we have a vector , then:

```math
\overrightarrow{AB} = a\cdot i + b\cdot j
```

where:

- $$a$$ is the vector component on the $$x$$-axis (horizontal)
- $$b$$ is the vector component on the $$y$$-axis (vertical)
- $$i$$ is the unit vector in the direction of the $$x$$-axis
- $$j$$ is the unit vector in the direction of the $$y$$-axis

Visible text: - is the vector component on the -axis (horizontal)
- is the vector component on the -axis (vertical)
- is the unit vector in the direction of the -axis
- is the unit vector in the direction of the -axis

### Example of Vector Components

Consider the vector $$\overrightarrow{AB}$$ in the figure. This vector can be written as:

Visible text: Consider the vector in the figure. This vector can be written as:

```math
\overrightarrow{AB} = 6i + 8j
```

This means that vector $$\overrightarrow{AB}$$ has a horizontal component of $$6 \text{ units}$$ to the right and a vertical component of $$8 \text{ units}$$ upward.

Visible text: This means that vector has a horizontal component of to the right and a vertical component of upward.

## Vector Magnitude from Its Components

When we know the components of a vector, we can calculate the length or magnitude of the vector using the Pythagorean theorem.

The magnitude of vector $$\overrightarrow{AB}$$ is denoted by $$|\overrightarrow{AB}|$$ and calculated using the formula:

Visible text: The magnitude of vector is denoted by and calculated using the formula:

```math
|\overrightarrow{AB}| = \sqrt{a^2 + b^2}
```

where $$a$$ and $$b$$ are the components of the vector.

Visible text: where and are the components of the vector.

### Example of Vector Magnitude Calculation

For the vector $$\overrightarrow{AB} = 6i + 8j$$, its magnitude is:

Visible text: For the vector , its magnitude is:

```math
|\overrightarrow{AB}| = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10
```

Therefore, the magnitude of vector $$\overrightarrow{AB}$$ is $$10 \text{ units}$$.

Visible text: Therefore, the magnitude of vector is .

## Vectors in Three-Dimensional Space

Vectors are not limited to a plane (two dimensions) but can also be extended to three-dimensional space.

Component: Vector3d
Props:
- title: Vectors in 3D Space
- description: Visualization of a vector and its components in three-dimensional space.
- vectors: [
{
from: [0, 0, 0],
to: [5, 7, 4],
color: getColor("ORANGE"),
label: "3D Vector",
},
{
from: [0, 0, 0],
to: [5, 0, 0],
color: getColor("TEAL"),
label: "x component",
},
{
from: [0, 0, 0],
to: [0, 7, 0],
color: getColor("LIME"),
label: "y component",
},
{
from: [0, 0, 0],
to: [0, 0, 4],
color: getColor("YELLOW"),
label: "z component",
},
]
- cameraPosition: [12, 8, 12]

In three-dimensional space, a vector has three components: the x-component, y-component, and z-component. A vector in three-dimensional space can be expressed as:

```math
\overrightarrow{v} = ai + bj + ck
```

where:

- $$a$$ is the vector component on the $$x$$-axis
- $$b$$ is the vector component on the $$y$$-axis
- $$c$$ is the vector component on the $$z$$-axis
- $$i$$, $$j$$, and $$k$$ are
  unit vectors in the direction of the x, y, and z axes

Visible text: - is the vector component on the -axis
- is the vector component on the -axis
- is the vector component on the -axis
- , , and are
 unit vectors in the direction of the x, y, and z axes

The magnitude of a vector in three-dimensional space is calculated using the formula:

```math
|\overrightarrow{v}| = \sqrt{a^2 + b^2 + c^2}
```

## Unit Direction Vector

To determine the direction of a vector, we can use a unit vector. A unit vector is a vector with a magnitude of $$1 \text{ unit}$$. To obtain a unit vector from a vector, we divide the vector by its magnitude.

Visible text: To determine the direction of a vector, we can use a unit vector. A unit vector is a vector with a magnitude of . To obtain a unit vector from a vector, we divide the vector by its magnitude.

The unit direction vector of $$\overrightarrow{AB}$$ is denoted by $$\hat{AB}$$ and calculated using:

Visible text: The unit direction vector of is denoted by and calculated using:

```math
\hat{AB} = \frac{\overrightarrow{AB}}{|\overrightarrow{AB}|}
```

### Example of Unit Direction Vector

For the vector $$\overrightarrow{AB} = 6i + 8j$$ with a magnitude of $$10$$, its unit direction vector is:

Visible text: For the vector with a magnitude of , its unit direction vector is:

```math
\hat{AB} = \frac{6i + 8j}{10} = \frac{6}{10}i + \frac{8}{10}j = 0.6i + 0.8j
```

This unit vector indicates the direction of vector $$\overrightarrow{AB}$$ without regard to its magnitude.

Visible text: This unit vector indicates the direction of vector without regard to its magnitude.

## Applications of Vector Components

Vector components have many applications in everyday life, such as:

- Calculating velocity and displacement in physics
- Analyzing forces in mechanics
- Determining the direction and magnitude of resultants in object movement
- Navigation and position determination in coordinate systems

By understanding vector components, we can analyze various problems involving direction and magnitude in mathematics and other applied sciences.