# Nakafa Learning Content

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

Learn equal vectors, negative vectors, zero vectors, and how scalar multiplication changes vector magnitude and direction.

---

## Equal Vectors Do Not Depend on Location

Two vectors are equal if they have the same magnitude and the same direction. Their positions in a diagram may be different. This matters because forces, displacements, and velocities are often drawn at different places while still representing the same vector.

> Two vectors are equal because their lengths and directions match, not because their tails sit at the same point.

Component: Vector3d
Props:
- title: Equal and Opposite Vectors
- description: Vectors $$\vec{a}$$ and $$\vec{b}$$ are
equal because they are parallel, point the same way, and have the same
length. Vector $$-\vec{a}$$ points the opposite way.
  Visible text: Vectors and are
equal because they are parallel, point the same way, and have the same
length. Vector points the opposite way.
- vectors: (() => {
const a = [4, 2, 0];
const bShift = [0, 0, 2];
const negativeShift = [0, 0, -2];
const b = a.map((component, index) => component + bShift[index]);
const negativeA = a.map(
(component, index) => -component + negativeShift[index]
);

return [
{
from: [0, 0, 0],
to: a,
color: getColor("TEAL"),
label: "a",
labelAnchorX: "center",
labelProgress: 3 / 5,
labelOffset: [0.15, 0.45, 0],
},
{
from: bShift,
to: b,
color: getColor("TEAL"),
label: "b",
labelAnchorX: "center",
labelProgress: 3 / 5,
labelOffset: [0.15, 0.45, 0.15],
},
{
from: negativeShift,
to: negativeA,
color: getColor("VIOLET"),
label: "-a",
labelAnchorX: "center",
labelProgress: 1 / 2,
labelOffset: [0.1, -0.55, 0],
},
];
})()
- cameraPosition: [8, 6, 10]
- cameraTarget: [0, -1.8, 0]

In the visual, $$\vec{a}$$ and $$\vec{b}$$ are not drawn on the same line, but they are still equal as vectors. Moving the drawing does not change the vector's magnitude or direction.

Visible text: In the visual, and are not drawn on the same line, but they are still equal as vectors. Moving the drawing does not change the vector's magnitude or direction.

## A Negative Vector Reverses Direction

The negative of $$\vec{a}$$ is written as $$-\vec{a}$$. It has the same magnitude as $$\vec{a}$$, but points in the opposite direction.

Visible text: The negative of is written as . It has the same magnitude as , but points in the opposite direction.

Component: MathContainer
Children:

```math
\lvert-\vec{a}\rvert=\lvert\vec{a}\rvert
```

```math
\vec{a}+(-\vec{a})=\vec{0}
```

For example, a $$12\text{ N}$$ force to the right and a $$12\text{ N}$$ force to the left form an opposite-vector pair. If both act on the same object along the same line, their resultant can be the zero vector.

Visible text: For example, a force to the right and a force to the left form an opposite-vector pair. If both act on the same object along the same line, their resultant can be the zero vector.

## Scalar Multiplication Changes Length

A scalar is an ordinary number with no direction. When a vector is multiplied by a scalar $$k$$, the vector length changes by a factor of $$\lvert k\rvert$$.

Visible text: A scalar is an ordinary number with no direction. When a vector is multiplied by a scalar , the vector length changes by a factor of .

| Form | Result magnitude | Result direction |
| :--- | :--------------- | :--------------- |
| $$2\vec{a}$$ | twice $$\vec{a}$$ | same as $$\vec{a}$$ |
| $$\frac{1}{2}\vec{a}$$ | half of $$\vec{a}$$ | same as $$\vec{a}$$ |
| $$-3\vec{a}$$ | three times $$\vec{a}$$ | opposite to $$\vec{a}$$ |
| $$0\vec{a}$$ | $$0$$ | becomes the zero vector |

Visible text: | Form | Result magnitude | Result direction |
| :--- | :--------------- | :--------------- |
| | twice | same as |
| | half of | same as |
| | three times | opposite to |
| | | becomes the zero vector |

A negative scalar does not make the magnitude negative. It reverses the vector direction.

## The Zero Vector as Cancellation

The zero vector is written as $$\vec{0}$$. Its magnitude is $$0$$, and its direction is undefined because the tail and tip coincide.

Visible text: The zero vector is written as . Its magnitude is , and its direction is undefined because the tail and tip coincide.

The zero vector often appears when vectors cancel each other. For example, two students may pull a box with equal forces in opposite directions. If the forces are exactly along the same line, their combination does not produce a net push to the right or to the left.

```math
\vec{F}_1+\vec{F}_2=\vec{0}
```

This idea becomes important in equilibrium: an object can remain at rest not because no forces act on it, but because the resultant of all forces is zero.