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

Learn zero, negative, and equivalent vectors, including their magnitude, direction, properties, and example exercises.

---

## Vector Magnitude and Direction

Every vector has two main components: **magnitude (length)** and **direction**.

Consider the example vector $$\overrightarrow{CD}$$ below.

Visible text: Consider the example vector below.

Component: VectorChart
Props:
- title: Vector $$CD$$
  Visible text: Vector
- description: Vector $$CD$$ with a length of $$4 \text{ cm}$$ and a direction of{" "}
$$45^\circ$$ relative to the horizontal line.
  Visible text: Vector with a length of and a direction of{" "}
 relative to the horizontal line.
- vectors: [
{
id: "cd",
name: "CD",
points: [
{ x: 0, y: 0 },
{ x: 4 * Math.cos(Math.PI / 4), y: 4 * Math.sin(Math.PI / 4) },
],
},
]

- The **magnitude of vector** $$\overrightarrow{CD}$$ is $$4 \text{ cm}$$. This indicates the _size_ or _value_ of the vector. The magnitude of a vector $$\vec{v}$$ is usually denoted as $$|\vec{v}|$$. So, $$|\overrightarrow{CD}| = 4 \text{ cm}$$.
- The **direction of vector** $$\overrightarrow{CD}$$ is $$45^\circ$$ relative to the horizontal line. This direction is crucial and distinguishes vectors from scalar quantities (which only have magnitude). Direction can be expressed using angles, compass points (like Northeast), or other references.

Visible text: - The **magnitude of vector** is . This indicates the _size_ or _value_ of the vector. The magnitude of a vector is usually denoted as . So, .
- The **direction of vector** is relative to the horizontal line. This direction is crucial and distinguishes vectors from scalar quantities (which only have magnitude). Direction can be expressed using angles, compass points (like Northeast), or other references.

## Negative Vector (Opposite Vector)

A negative vector or opposite vector is a vector that has the **same magnitude** but **opposite direction** to the original vector.

Imagine Andi walks $$100 \text{ m}$$ in a direction of $$30^\circ$$ (let's call this displacement vector $$\vec{A}$$). Then, Andi returns to the starting position. This second displacement is the opposite vector of $$\vec{A}$$, which we write as $$-\vec{A}$$.

Visible text: Imagine Andi walks in a direction of (let's call this displacement vector ). Then, Andi returns to the starting position. This second displacement is the opposite vector of , which we write as .

Component: VectorChart
Props:
- title: Vector and Opposite Vector
- description: Vector $$A$$ and vector $$-A$$ have
the same magnitude but opposite directions.
  Visible text: Vector and vector have
the same magnitude but opposite directions.
- vectors: [
{
id: "A",
name: "A",
points: [
{ x: 0, y: 0 },
{ x: 5, y: 2 },
],
},
{
id: "-A",
name: "-A",
points: [
{ x: 0, y: 0.5 },
{ x: 5, y: 2.5 },
],
direction: "backward",
},
]

- Vectors $$\vec{A}$$ and $$-\vec{A}$$ have the same magnitude ($$|\vec{A}| = |-\vec{A}|$$).
- The direction of vector $$-\vec{A}$$ is exactly opposite to the direction of vector $$\vec{A}$$. If $$\vec{A}$$ points in one direction, $$-\vec{A}$$ points in the opposite direction ($$180^\circ$$ difference).

Visible text: - Vectors and have the same magnitude ().
- The direction of vector is exactly opposite to the direction of vector . If points in one direction, points in the opposite direction ( difference).

## Zero Vector

The zero vector is a special vector because it has **zero magnitude**. Due to its zero length, this vector **does not have a specific direction**.

The zero vector can be visualized as a single point, where the initial point and terminal point coincide. The zero vector is usually denoted by $$\vec{0}$$.

Visible text: The zero vector can be visualized as a single point, where the initial point and terminal point coincide. The zero vector is usually denoted by .

**Example**:

If Andi walks $$100 \text{ m}$$ east, then walks back $$100 \text{ m}$$ west, Andi's total displacement is zero. This total displacement can be represented as the zero vector ($$\vec{0}$$).

Visible text: If Andi walks east, then walks back west, Andi's total displacement is zero. This total displacement can be represented as the zero vector ().

## Equivalent Vectors (Equal Vectors)

Two or more vectors are said to be **equivalent** or **equal** if they have the **same magnitude (length) and direction**, even if their starting points are different.

Consider the graph below showing three equivalent vectors: $$\overrightarrow{CD}$$, $$\overrightarrow{EF}$$, and $$\overrightarrow{KL}$$.

Visible text: Consider the graph below showing three equivalent vectors: , , and .

Component: VectorChart
Props:
- title: Equivalent Vectors
- description: Vectors $$CD$$, $$EF$$, and{" "}
$$KL$$ have the same magnitude and direction.
  Visible text: Vectors , , and{" "}
 have the same magnitude and direction.
- vectors: [
{
id: "CD",
name: "CD",
points: [
{ x: 2, y: 1 }, // C
{ x: 5, y: 3 }, // D
],
},
{
id: "EF",
name: "EF",
points: [
{ x: 4, y: 2 }, // E
{ x: 7, y: 4 }, // F
],
},
{
id: "KL",
name: "KL",
points: [
{ x: 3, y: 0 }, // K
{ x: 6, y: 2 }, // L
],
},
]

The three vectors above have the same magnitude and direction, so they are equivalent. We can write this as:

```math
\overrightarrow{CD} = \overrightarrow{EF} = \overrightarrow{KL}
```

A vector is said to be equivalent to another vector if it has the same magnitude and direction as the other vector.

## Exercise

Consider the two vectors below:

Component: VectorChart
Props:
- title: Comparison of Vector $$A$$ and Vector{" "}
$$B$$
  Visible text: Comparison of Vector and Vector{" "}
- description: Comparison of Vector $$A$$ and Vector{" "}
$$B$$
  Visible text: Comparison of Vector and Vector{" "}
- vectors: [
{
id: "A",
name: "A",
points: [
{ x: 1, y: 4 },
{ x: 0, y: 0 },
],
direction: "backward",
},
{
id: "B",
name: "B",
points: [
{ x: 1, y: 4 },
{ x: 5, y: 5 },
],
},
]

Is vector $$\vec{A}$$ the opposite vector of $$\vec{B}$$?

Visible text: Is vector the opposite vector of ?

**Answer:**

Vector $$\vec{A}$$ is **not** the opposite vector of $$\vec{B}$$. To be an opposite vector, two conditions must be met:

Visible text: Vector is **not** the opposite vector of . To be an opposite vector, two conditions must be met:

1.  **They must have the same magnitude:** Visually, $$|\vec{A}| \neq |\vec{B}|$$ (their lengths appear different).
2.  **Their directions must be exactly opposite ($$180^\circ$$ apart):** The direction of $$\vec{A}$$ is not opposite to the direction of $$\vec{B}$$.

Visible text: 1. **They must have the same magnitude:** Visually, (their lengths appear different).
2. **Their directions must be exactly opposite ( apart):** The direction of is not opposite to the direction of .

Since these two conditions are not met, $$\vec{A}$$ is not the opposite vector of $$\vec{B}$$.

Visible text: Since these two conditions are not met, is not the opposite vector of .

**How to make vectors $$\vec{A}$$ and $$\vec{B}$$ opposite?**

Visible text: **How to make vectors and opposite?**

To make vectors $$\vec{A}$$ and $$\vec{B}$$ opposite, you could define them like this:

Visible text: To make vectors and opposite, you could define them like this:

Component: VectorChart
Props:
- title: Opposite Vectors $$A$$ and $$B$$
  Visible text: Opposite Vectors and
- description: Vector $$A$$ and $$B$$ are opposite
  Visible text: Vector and are opposite
- vectors: [
{
id: "A",
name: "A",
points: [
{ x: 1, y: 4 },
{ x: 0, y: 0 },
],
direction: "backward",
},
{
id: "B",
name: "B",
points: [
{ x: 1, y: 4 },
{ x: 4, y: 8 },
],
direction: "forward",
},
]