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URL: https://nakafa.com/en/subjects/mathematics/vector-operations/vector-concept
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Learn what vectors are through examples like wind maps, then study magnitude, direction, notation, and vector properties.

---

## What Is a Vector?

To understand vectors, imagine a weather map showing wind direction and speed at various locations. On such maps, arrows are often drawn. These arrows indicate the **direction** of the wind and **how strong** it blows (usually shown by the length or thickness of the arrow).

In mathematics, quantities that have both **magnitude (size)** and **direction** like this are called **vectors**. Thus, the speed and direction of wind on a weather map is a good example of a vector. The length of the arrow represents the _magnitude_ of the wind speed (for example, $$30 \text{ km/h}$$), and the direction of the arrow shows the _direction_ in which it blows.

Visible text: In mathematics, quantities that have both **magnitude (size)** and **direction** like this are called **vectors**. Thus, the speed and direction of wind on a weather map is a good example of a vector. The length of the arrow represents the _magnitude_ of the wind speed (for example, ), and the direction of the arrow shows the _direction_ in which it blows.

## Drawing and Writing Vectors

Vectors are typically drawn as directed line segments (arrows).

Component: VectorChart
Props:
- title: Vector Graph
- description: Vectors in a Cartesian coordinate system with horizontal{" "}
$$x$$ and vertical $$y$$.
  Visible text: Vectors in a Cartesian coordinate system with horizontal{" "}
 and vertical .
- vectors: [
{
id: "vector_1",
name: "Vector 1",
points: [
{ x: 0, y: 0 },
{ x: 6, y: 8 },
],
},
{
id: "vector_2",
name: "Vector 2",
points: [
{ x: 1, y: 1 },
{ x: 5, y: 2 },
],
direction: "backward",
},
{
id: "vector_3",
name: "Vector 3",
points: [
{ x: 2, y: 0 },
{ x: 4, y: 7 },
],
direction: "backward",
},
]

- **Vector** $$1$$: Direction of the vector from point $$(0, 0)$$ to point $$(6, 8)$$
- **Vector** $$2$$: Direction of the vector from point $$(5, 2)$$ to point $$(1, 1)$$
- **Vector** $$3$$: Direction of the vector from point $$(4, 7)$$ to point $$(2, 0)$$

Visible text: - **Vector** : Direction of the vector from point to point 
- **Vector** : Direction of the vector from point to point 
- **Vector** : Direction of the vector from point to point

### Key Components of Vectors

1.  **Initial Point:** The starting point of the vector.
2.  **Terminal Point:** The ending point of the vector, marked with an arrowhead.
3.  **Magnitude:** The length of the arrow, representing the value or size of the vector. Often also called _vector length_. The length of vector $$\overrightarrow{AB}$$ is denoted as $$|\overrightarrow{AB}|$$.
4.  **Direction:** The direction pointed by the arrowhead.

Visible text: 1. **Initial Point:** The starting point of the vector.
2. **Terminal Point:** The ending point of the vector, marked with an arrowhead.
3. **Magnitude:** The length of the arrow, representing the value or size of the vector. Often also called _vector length_. The length of vector is denoted as .
4. **Direction:** The direction pointed by the arrowhead.

### Vector Notation

There are several ways to write vectors:

1.  **Two Capital Letters with an Arrow Above:** Indicating the initial and terminal points. Example: $$\overrightarrow{AB}$$, meaning a vector from point $$A$$ to point $$B$$.
2.  **One Lowercase Letter with an Arrow Above:** Example: $$\vec{a}$$.
3.  **Bold Letters:** Example: **a** or **AB**

Visible text: 1. **Two Capital Letters with an Arrow Above:** Indicating the initial and terminal points. Example: , meaning a vector from point to point .
2. **One Lowercase Letter with an Arrow Above:** Example: .
3. **Bold Letters:** Example: **a** or **AB**

## Exercise

Are the shapes below vectors? Why?

Component: VectorChart
Props:
- title: Cartesian Coordinate System
- description: Horizontal $$x$$
-axis and vertical $$y$$
-axis.
  Visible text: Horizontal 
-axis and vertical 
-axis.
- vectors: [
{
id: "shape_1",
name: "Shape 1",
points: [
{ x: 0, y: 0 },
{ x: 1, y: 9 },
{ x: 3, y: 4 },
{ x: 6, y: 8 },
],
},
{
id: "shape_2",
name: "Shape 2",
points: [
{ x: 1, y: 1 },
{ x: 2, y: 6 },
{ x: 5, y: 2 },
],
direction: "backward",
},
{
id: "shape_3",
name: "Shape 3",
points: [
{ x: 2, y: 0 },
{ x: 4, y: 7 },
],
direction: "both",
},
]

These shapes are **not vectors** because:

1. **Shape** $$1$$: because its line is curved, whereas vectors must be represented by straight lines.
2. **Shape** $$2$$: because it consists of more than one straight line, whereas a vector must be represented by a single straight line.
3. **Shape** $$3$$: because it has two directions (shown by arrows at both ends), whereas a vector must have only one clear direction.

Visible text: 1. **Shape** : because its line is curved, whereas vectors must be represented by straight lines.
2. **Shape** : because it consists of more than one straight line, whereas a vector must be represented by a single straight line.
3. **Shape** : because it has two directions (shown by arrows at both ends), whereas a vector must have only one clear direction.

A valid vector in mathematics must have these characteristics:

- It must be a straight line
- It must have one direction indicated by an arrow
- It must consist of a single straight line (not broken or curved)