# Nakafa Learning Content

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URL: https://nakafa.com/en/subjects/physics/vector/component
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/material/lesson/physics/vector/component/en.mdx

Learn how to express a vector as components along the x, y, and z axes using unit vectors.

---

## Why Vectors Are Split into Components

A slanted vector is difficult to calculate with if we only see it as one arrow. To make calculations easier, the vector is split into parts parallel to the coordinate axes. These parts are called **vector components**.

The horizontal component shows how much of the vector lies along the $$x$$ axis. The vertical component shows how much lies along the $$y$$ axis. In three-dimensional space, there is also a component along the $$z$$ axis.

Visible text: The horizontal component shows how much of the vector lies along the axis. The vertical component shows how much lies along the axis. In three-dimensional space, there is also a component along the axis.

> Vector components turn one slanted arrow into parts that follow the axes.

## Reading Components in a Plane

In a plane, vector $$\vec{A}$$ can be written as the combination of its $$x$$ and $$y$$ components.

Visible text: In a plane, vector can be written as the combination of its and components.

```math
\vec{A}=A_x\hat{i}+A_y\hat{j}
```

The notation $$\hat{i}$$ is the unit vector in the $$x$$ direction, while $$\hat{j}$$ is the unit vector in the $$y$$ direction. The numbers $$A_x$$ and $$A_y$$ give the component sizes along each direction.

Visible text: The notation is the unit vector in the direction, while is the unit vector in the direction. The numbers and give the component sizes along each direction.

For example, $$\vec{A}=6\hat{i}+8\hat{j}$$ means the vector moves $$6$$ units in the positive $$x$$ direction and $$8$$ units in the positive $$y$$ direction.

Visible text: For example, means the vector moves units in the positive direction and units in the positive direction.

```math
\lvert\vec{A}\rvert=\sqrt{6^2+8^2}=10
```

The vector length comes from the Pythagorean theorem because the $$x$$ and $$y$$ components are perpendicular.

Visible text: The vector length comes from the Pythagorean theorem because the and components are perpendicular.

## Components in Three-Dimensional Space

In three-dimensional space, a vector needs three components. The general form is:

```math
\vec{A}=A_x\hat{i}+A_y\hat{j}+A_z\hat{k}
```

The unit vector $$\hat{k}$$ points along the $$z$$ axis. This component is needed when motion or force is not limited to a flat plane, such as a drone moving east, north, and upward at the same time.

Visible text: The unit vector points along the axis. This component is needed when motion or force is not limited to a flat plane, such as a drone moving east, north, and upward at the same time.

Component: Vector3d
Props:
- title: Components of $$\vec{A}$$
  Visible text: Components of
- description: Vector $$\vec{A}=4\hat{i}+3\hat{j}+2\hat{k}$$ is split
into three perpendicular components.
  Visible text: Vector is split
into three perpendicular components.
- vectors: (() => {
const components = [
[4, 0, 0],
[0, 3, 0],
[0, 0, 2],
];
const points = components.reduce(
(path, component) => [
...path,
path.at(-1).map((value, index) => value + component[index]),
],
[[0, 0, 0]]
);
const resultant = components.reduce(
(sum, component) =>
sum.map((value, index) => value + component[index]),
[0, 0, 0]
);

return [
{
from: [0, 0, 0],
to: resultant,
color: getColor("VIOLET"),
label: "A",
labelAnchorX: "center",
labelProgress: 3 / 5,
labelOffset: [0.35, 0.55, 0.3],
},
{
from: points[0],
to: points[1],
color: getColor("TEAL"),
label: "Ax",
labelAnchorX: "center",
labelProgress: 1 / 2,
labelOffset: [0, -0.45, 0],
},
{
from: points[1],
to: points[2],
color: getColor("AMBER"),
label: "Ay",
labelAnchorX: "center",
labelProgress: 1 / 2,
labelOffset: [0.55, 0, 0.15],
},
{
from: points[2],
to: points[3],
color: getColor("CYAN"),
label: "Az",
labelAnchorX: "center",
labelProgress: 1 / 2,
labelOffset: [0.65, -0.25, 0.15],
},
];
})()
- cameraPosition: [8, 6, 9]
- cameraTarget: [2, -0.5, 1]

The visual uses component steps in order: move along the $$x$$ axis, then the $$y$$ axis, then the $$z$$ axis. The final endpoint is the same as the endpoint of vector $$\vec{A}$$.

Visible text: The visual uses component steps in order: move along the axis, then the axis, then the axis. The final endpoint is the same as the endpoint of vector .

## Positive and Negative Component Signs

Components can be positive or negative. The sign shows direction relative to the reference axis, not a negative vector magnitude.

| Component | Positive sign means | Negative sign means |
| :-------- | :------------------ | :------------------ |
| $$A_x$$ | toward positive $$x$$ | toward negative $$x$$ |
| $$A_y$$ | toward positive $$y$$ | toward negative $$y$$ |
| $$A_z$$ | toward positive $$z$$ | toward negative $$z$$ |

Visible text: | Component | Positive sign means | Negative sign means |
| :-------- | :------------------ | :------------------ |
| | toward positive | toward negative |
| | toward positive | toward negative |
| | toward positive | toward negative |

If $$\vec{B}=-5\hat{i}+2\hat{j}$$, the vector has a $$5$$ unit component toward negative $$x$$ and a $$2$$ unit component toward positive $$y$$.

Visible text: If , the vector has a unit component toward negative and a unit component toward positive .

## Displacement Written as Components

A student walks $$3\text{ m}$$ east, then $$4\text{ m}$$ north. If east is the positive $$x$$ direction and north is the positive $$y$$ direction, the displacement is:

Visible text: A student walks east, then north. If east is the positive direction and north is the positive direction, the displacement is:

Component: MathContainer
Children:

```math
\vec{s}=3\hat{i}+4\hat{j}
```

```math
\lvert\vec{s}\rvert=\sqrt{3^2+4^2}=5\text{ m}
```

Components keep the calculation organized because each direction is handled on its own axis.