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

Learn equivalent vectors with same magnitude and direction. Learn properties, component representation, and real-world applications in physics and engineering.

---

## Definition of Equivalent Vectors

Two vectors are said to be equivalent if they have the same magnitude (length) and direction. Mathematically, two vectors $$\vec{a}$$ and $$\vec{b}$$ are equivalent if their components are equal. In mathematical notation, this can be written as $$\vec{a} \equiv \vec{b}$$.

Visible text: Two vectors are said to be equivalent if they have the same magnitude (length) and direction. Mathematically, two vectors and are equivalent if their components are equal. In mathematical notation, this can be written as .

Equivalent vectors can have different positions in a plane or space, but they maintain the same magnitude and direction.

Component: Vector3d
Props:
- title: Example of Equivalent Vectors
- description: The purple and orange vectors have the same magnitude and direction, making them equivalent despite their different positions.
- vectors: [
{
from: [0, 0, 0],
to: [3, 2, 1],
color: getColor("VIOLET"),
label: "v₁",
},
{
from: [1, 1, 1],
to: [4, 3, 2],
color: getColor("ORANGE"),
label: "v₂",
},
]
- cameraPosition: [8, 5, 8]

## Conditions for Equivalent Vectors

Two vectors $$\overrightarrow{CD}$$ and $$\overrightarrow{PQ}$$ are said to be equivalent if:

Visible text: Two vectors and are said to be equivalent if:

1. Both vectors have equal length: $$|\overrightarrow{CD}| = |\overrightarrow{PQ}|$$
2. Both vectors have the same direction

Visible text: 1. Both vectors have equal length: 
2. Both vectors have the same direction

## Representation of Equivalent Vectors

### In Component Form

In a two-dimensional Cartesian plane, two vectors $$\vec{a}$$ and $$\vec{b}$$ are equivalent if:

Visible text: In a two-dimensional Cartesian plane, two vectors and are equivalent if:

```math
\vec{a} = (a_1, a_2) \equiv \vec{b} = (b_1, b_2)
```

where $$a_1 = b_1$$ and $$a_2 = b_2$$

Visible text: where and

In three-dimensional space, vectors $$\vec{a}$$ and $$\vec{b}$$ are equivalent if:

Visible text: In three-dimensional space, vectors and are equivalent if:

```math
\vec{a} = (a_1, a_2, a_3) \equiv \vec{b} = (b_1, b_2, b_3)
```

where $$a_1 = b_1$$, $$a_2 = b_2$$, and $$a_3 = b_3$$

Visible text: where , , and

Component: Vector3d
Props:
- title: Equivalent Vectors in Component Form
- description: Two vectors with the same components are always equivalent, even if they have different positions in space.
- vectors: [
{
from: [0, 0, 0],
to: [2, 3, 1],
color: getColor("FUCHSIA"),
label: "a = (2, 3, 1)",
},
{
from: [3, 0, 2],
to: [5, 3, 3],
color: getColor("CYAN"),
label: "b = (2, 3, 1)",
},
]

### In Terms of Initial and Terminal Points

If vector $$\overrightarrow{AB}$$ has initial point $$A(x_1, y_1)$$ and terminal point $$B(x_2, y_2)$$, then the vector can be expressed as:

Visible text: If vector has initial point and terminal point , then the vector can be expressed as:

```math
\overrightarrow{AB} = (x_2 - x_1, y_2 - y_1)
```

Two vectors $$\overrightarrow{AB}$$ and $$\overrightarrow{CD}$$ are equivalent if:

Visible text: Two vectors and are equivalent if:

```math
(x_2 - x_1, y_2 - y_1) = (x_4 - x_3, y_4 - y_3)
```

where $$C(x_3, y_3)$$ and $$D(x_4, y_4)$$

Visible text: where and

## Properties of Equivalent Vectors

### Reflexive Property

Every vector is equivalent to itself.

```math
\vec{a} \equiv \vec{a}
```

### Symmetric Property

If vector $$\vec{a}$$ is equivalent to vector $$\vec{b}$$, then vector $$\vec{b}$$ is also equivalent to vector $$\vec{a}$$.

Visible text: If vector is equivalent to vector , then vector is also equivalent to vector .

```math
\text{If } \vec{a} \equiv \vec{b} \text{, then } \vec{b} \equiv \vec{a}
```

### Transitive Property

If vector $$\vec{a}$$ is equivalent to vector $$\vec{b}$$ and vector $$\vec{b}$$ is equivalent to vector $$\vec{c}$$, then vector $$\vec{a}$$ is equivalent to vector $$\vec{c}$$.

Visible text: If vector is equivalent to vector and vector is equivalent to vector , then vector is equivalent to vector .

Component: ContentStack
Children:

```math
\text{If } \vec{a} \equiv \vec{b} \text{ and} \vec{b} \equiv \vec{c} \text{, then } \vec{a} \equiv \vec{c}
```

Component: Vector3d
Props:
- title: Transitive Property of Equivalent Vectors
- description: Three equivalent vectors: if $$a \equiv b$$ and{" "}
$$b \equiv c$$, then $$a \equiv c$$.
  Visible text: Three equivalent vectors: if and{" "}
, then .
- vectors: [
{
from: [0, 0, 0],
to: [2, 2, 0],
color: getColor("AMBER"),
label: "a",
},
{
from: [1, 1, 2],
to: [3, 3, 2],
color: getColor("EMERALD"),
label: "b",
},
{
from: [2, 0, 1],
to: [4, 2, 1],
color: getColor("PINK"),
label: "c",
},
]
- cameraPosition: [7, 5, 7]

## Examples of Equivalent Vectors

### First Example

Vector $$\overrightarrow{AB}$$ with $$A(2, 3)$$ and $$B(5, 7)$$ is equivalent to vector $$\overrightarrow{CD}$$ with $$C(1, 1)$$ and $$D(4, 5)$$.

Visible text: Vector with and is equivalent to vector with and .

Proof:

Component: MathContainer
Children:

```math
\overrightarrow{AB} = (5-2, 7-3) = (3, 4)
```

```math
\overrightarrow{CD} = (4-1, 5-1) = (3, 4)
```

Since $$\overrightarrow{AB} = \overrightarrow{CD} = (3, 4)$$, vector $$\overrightarrow{AB}$$ is equivalent to vector $$\overrightarrow{CD}$$.

Visible text: Since , vector is equivalent to vector .

Component: Vector3d
Props:
- title: Vectors $$AB$$ and $$CD$$
  Visible text: Vectors and
- description: Visualization of vectors $$AB(3,4,0)$$ and{" "}
$$CD(3,4,0)$$ which are equivalent in space.
  Visible text: Visualization of vectors and{" "}
 which are equivalent in space.
- vectors: [
{
from: [2, 3, 0],
to: [5, 7, 0],
color: getColor("VIOLET"),
label: "AB",
},
{
from: [1, 1, 0],
to: [4, 5, 0],
color: getColor("YELLOW"),
label: "CD",
},
]
- cameraPosition: [10, 10, 10]

### Second Example

Vector $$\overrightarrow{PQ}$$ with $$P(0, 0)$$ and $$Q(2, 2)$$ is equivalent to vector $$\overrightarrow{RS}$$ with $$R(3, 1)$$ and $$S(5, 3)$$.

Visible text: Vector with and is equivalent to vector with and .

Proof:

Component: MathContainer
Children:

```math
\overrightarrow{PQ} = (2-0, 2-0) = (2, 2)
```

```math
\overrightarrow{RS} = (5-3, 3-1) = (2, 2)
```

Since $$\overrightarrow{PQ} = \overrightarrow{RS} = (2, 2)$$, vector $$\overrightarrow{PQ}$$ is equivalent to vector $$\overrightarrow{RS}$$.

Visible text: Since , vector is equivalent to vector .

Component: Vector3d
Props:
- title: Vectors $$PQ$$ and $$RS$$
  Visible text: Vectors and
- description: Visualization of vectors $$PQ(2,2,0)$$ and{" "}
$$RS(2,2,0)$$ which are equivalent in space.
  Visible text: Visualization of vectors and{" "}
 which are equivalent in space.
- vectors: [
{
from: [0, 0, 0],
to: [2, 2, 0],
color: getColor("TEAL"),
label: "PQ",
},
{
from: [3, 1, 0],
to: [5, 3, 0],
color: getColor("ROSE"),
label: "RS",
},
]
- cameraPosition: [8, 6, 8]

## Applications of Equivalent Vectors

The concept of equivalent vectors is important in various applications, including:

1. In physics, for calculating displacement, velocity, and acceleration of objects
2. In navigation, for determining direction and travel distance
3. In computer graphics, for object transformation
4. In electrical engineering, for representing magnetic and electric forces