# Nakafa Learning Content

> For AI agents: use [llms.txt](https://nakafa.com/llms.txt) for the site index. Markdown versions are available by appending `.md` to content URLs or sending `Accept: text/markdown`.

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

Learn two-dimensional vectors on coordinate planes. Learn vector components, magnitude calculations, and unit vectors with visual demonstrations.

---

## Two-Dimensional Vector Concept

In the Cartesian coordinate system, each point on a plane can be represented by a pair of numbers $$(x, y)$$, where $$x$$ is the horizontal position and $$y$$ is the vertical position. The origin point is $$O(0, 0)$$.

Visible text: In the Cartesian coordinate system, each point on a plane can be represented by a pair of numbers , where is the horizontal position and is the vertical position. The origin point is .

If we draw a straight line from the origin $$O$$ to another point, for example $$Q(x, y)$$, we get a **vector**. This vector is often written as $$\overrightarrow{OQ}$$. A vector has both **magnitude** (line length) and **direction** (indicated by the arrow).

Visible text: If we draw a straight line from the origin to another point, for example , we get a **vector**. This vector is often written as . A vector has both **magnitude** (line length) and **direction** (indicated by the arrow).

To simplify, we use **unit vectors**. Unit vectors have a length of $$1 \text{ unit}$$.

Visible text: To simplify, we use **unit vectors**. Unit vectors have a length of .

- $$\mathbf{i}$$ is the unit vector in the positive $$x$$
  -axis direction (horizontal).
- $$\mathbf{j}$$ is the unit vector in the positive $$y$$
  -axis direction (vertical).

Visible text: - is the unit vector in the positive 
 -axis direction (horizontal).
- is the unit vector in the positive 
 -axis direction (vertical).

Vector $$\overrightarrow{OQ}$$ can be expressed as a combination of horizontal movement of $$x$$ and vertical movement of $$y$$. In unit vector form, we write:

Visible text: Vector can be expressed as a combination of horizontal movement of and vertical movement of . In unit vector form, we write:

```math
\overrightarrow{OQ} = x\mathbf{i} + y\mathbf{j}
```

## Vector Components and Magnitude

The values $$x$$ and $$y$$ in vector $$\overrightarrow{OQ} = x\mathbf{i} + y\mathbf{j}$$ are called the vector **components**.

Visible text: The values and in vector are called the vector **components**.

- $$x$$ is the **horizontal component**. It's like the shadow of
  the vector on the $$x$$
  -axis when illuminated from above.
- $$y$$ is the **vertical component**. It's like the shadow of
  the vector on the $$y$$
  -axis when illuminated from the side.

Visible text: - is the **horizontal component**. It's like the shadow of
 the vector on the 
 -axis when illuminated from above.
- is the **vertical component**. It's like the shadow of
 the vector on the 
 -axis when illuminated from the side.

A vector with these two components is called a **two-dimensional vector**.

The **length** or **magnitude** of vector $$\overrightarrow{OQ}$$, written as $$|\overrightarrow{OQ}|$$, is the distance from the origin point $$O$$ to the endpoint $$Q$$. If $$Q(x,y)$$ is the endpoint of the vector and $$R(x,0)$$ is the projection of point $$Q$$ onto the $$x$$-axis, we can calculate it using the Pythagorean theorem on the right-angled triangle $$ORQ$$:

Visible text: The **length** or **magnitude** of vector , written as , is the distance from the origin point to the endpoint . If is the endpoint of the vector and is the projection of point onto the -axis, we can calculate it using the Pythagorean theorem on the right-angled triangle :

```math
|\overrightarrow{OQ}| = \sqrt{x^2 + y^2}
```

## Two-Dimensional Vector Visualization

Observe the following vector visualization to understand the concept of vectors in the Cartesian plane:

Component: Vector3d
Props:
- title: Vector in the Coordinate Plane
- description: Visualization of vector{" "}
$$OQ = 4\mathbf{i} + 3\mathbf{j}$$
in the Cartesian coordinate system
  Visible text: Visualization of vector{" "}

in the Cartesian coordinate system
- vectors: [
{
from: [0, 0, 0],
to: [4, 3, 0],
color: getColor("LIME"),
label: "Vector OQ",
},
{
from: [0, 0, 0],
to: [4, 0, 0],
color: getColor("TEAL"),
label: "X Component",
},
{
from: [0, 0, 0],
to: [0, 3, 0],
color: getColor("YELLOW"),
label: "Y Component",
},
]

In this visualization, we don't use the $$z$$-axis because we're working in a two-dimensional plane.

Visible text: In this visualization, we don't use the -axis because we're working in a two-dimensional plane.

In the visualization above:

- Vector $$\overrightarrow{OQ}$$ (light green) has initial point $$O(0,0)$$ and endpoint $$Q(4,3)$$
- The $$x$$-component (light blue) is the projection of vector $$\overrightarrow{OQ}$$ on the $$x$$-axis, which is $$4\mathbf{i}$$
- The $$y$$-component (yellow) is the projection of vector $$\overrightarrow{OQ}$$ on the $$y$$-axis, which is $$3\mathbf{j}$$
- The magnitude of the vector $$|\overrightarrow{OQ}| = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5$$

Visible text: - Vector (light green) has initial point and endpoint 
- The -component (light blue) is the projection of vector on the -axis, which is 
- The -component (yellow) is the projection of vector on the -axis, which is 
- The magnitude of the vector