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URL: https://nakafa.com/en/subjects/physics/vector/notation
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Learn how to read vector symbols, two-point notation, vector magnitude, zero vectors, and unit vectors in physics.

---

## Arrows as Vector Language

A vector is commonly drawn as an arrow. The drawing is not decorative. The tail shows the starting point, the tip shows the direction, and the arrow length represents the vector magnitude.

If a force arrow is longer, the force is larger on the same drawing scale. If the arrow points up and to the right, the force acts up and to the right.

> Vector notation should keep two pieces of information visible: magnitude and direction.

## Two-Point Notation

A vector can be written from a starting point to an ending point. For example, $$\overrightarrow{AB}$$ means the vector from point $$A$$ to point $$B$$.

Visible text: A vector can be written from a starting point to an ending point. For example, means the vector from point to point .

The order of the letters matters. $$\overrightarrow{AB}$$ and $$\overrightarrow{BA}$$ use the same two points, but their directions are opposite.

Visible text: The order of the letters matters. and use the same two points, but their directions are opposite.

| Notation | How to read it | Direction meaning |
| :------- | :------------- | :---------------- |
| $$\overrightarrow{AB}$$ | vector from $$A$$ to $$B$$ | tail at $$A$$, tip at $$B$$ |
| $$\overrightarrow{BA}$$ | vector from $$B$$ to $$A$$ | tail at $$B$$, tip at $$A$$ |
| $$\lvert\overrightarrow{AB}\rvert$$ | magnitude of $$\overrightarrow{AB}$$ | length only, not direction |

Visible text: | Notation | How to read it | Direction meaning |
| :------- | :------------- | :---------------- |
| | vector from to | tail at , tip at |
| | vector from to | tail at , tip at |
| | magnitude of | length only, not direction |

The arrow above the letters tells us that the quantity is a vector. Absolute-value bars tell us that only the magnitude is being discussed.

## Single-Letter Notation

In calculations, vectors are often named with one letter such as $$\vec{a}$$, $$\vec{v}$$, or $$\vec{F}$$. The letter helps identify the physical quantity.

Visible text: In calculations, vectors are often named with one letter such as , , or . The letter helps identify the physical quantity.

For example, $$\vec{F}$$ is often used for force, $$\vec{v}$$ for velocity, and $$\vec{s}$$ for displacement. The magnitude is written without the arrow, such as $$F$$ or $$\lvert\vec{F}\rvert$$.

Visible text: For example, is often used for force, for velocity, and for displacement. The magnitude is written without the arrow, such as or .

Component: MathContainer
Children:

```math
\vec{F}=20\text{ N to the right}
```

```math
\lvert\vec{F}\rvert=20\text{ N}
```

The first line still includes direction. The second line states only the force magnitude.

## Zero Vectors and Unit Vectors

Two special vector names appear often once vectors are calculated using components.

A **zero vector** has magnitude $$0$$. Its tail and tip coincide, so its direction is undefined. In physics, a zero vector can appear when two equal and opposite forces cancel each other.

Visible text: A **zero vector** has magnitude . Its tail and tip coincide, so its direction is undefined. In physics, a zero vector can appear when two equal and opposite forces cancel each other.

A **unit vector** has magnitude $$1$$ and only shows direction. Unit vectors do not tell us how strong a force is or how far a displacement is. They identify axis directions.

Visible text: A **unit vector** has magnitude and only shows direction. Unit vectors do not tell us how strong a force is or how far a displacement is. They identify axis directions.

Component: MathContainer
Children:

```math
\hat{i}=\text{unit vector in the }x\text{ direction}
```

```math
\hat{j}=\text{unit vector in the }y\text{ direction}
```

```math
\hat{k}=\text{unit vector in the }z\text{ direction}
```

Later, when a vector is written as $$\vec{A}=A_x\hat{i}+A_y\hat{j}$$, $$\hat{i}$$ and $$\hat{j}$$ give the directions, while $$A_x$$ and $$A_y$$ give the component magnitudes along each axis.

Visible text: Later, when a vector is written as , and give the directions, while and give the component magnitudes along each axis.

## Common Reading Mistakes

The first mistake is treating $$\overrightarrow{AB}$$ as the same vector as $$\overrightarrow{BA}$$ because they use the same two points. Their directions are opposite.

Visible text: The first mistake is treating as the same vector as because they use the same two points. Their directions are opposite.

The second mistake is writing a vector magnitude as a negative number. A magnitude is always $$0$$ or positive. A negative sign on a vector usually means the direction is opposite to a chosen reference direction, not that the length is negative.

Visible text: The second mistake is writing a vector magnitude as a negative number. A magnitude is always or positive. A negative sign on a vector usually means the direction is opposite to a chosen reference direction, not that the length is negative.

For example, $$-\vec{a}$$ does not mean the vector has negative magnitude. It means the vector has the same magnitude as $$\vec{a}$$ but points in the opposite direction.

Visible text: For example, does not mean the vector has negative magnitude. It means the vector has the same magnitude as but points in the opposite direction.