Vector Symbols and Notation

What a Vector Arrow Shows

A vector is commonly drawn as an arrow. Its tail shows the starting point, its tip shows the direction, and its 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.

A complete vector notation shows both 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 $B$ .

The order of the letters matters. $\overrightarrow{AB}$ and $\overrightarrow{BA}$ 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 $\overrightarrow{AB}$	Gives the vector's length. Direction does not affect this value

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.

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 $\lvert\vec{F}\rvert$ .

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

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.

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. In the standard three-dimensional Cartesian basis, each unit vector has one component equal to $1$ :

\hat{i}=(1,0,0)

Later, when a vector is written as $\vec{A}=A_x\hat{i}+A_y\hat{j}$ , $\hat{i}$ and $\hat{j}$ give the positive axis directions, while the signed scalars $A_x$ and $A_y$ give the components along those axes.

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.

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 points the vector opposite to the chosen reference direction. It does not make the length 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.