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1.2.2. Vector Notation

Table of Contents

Symbols and Meaning

Vector notation is the written language we use to tell vectors apart from ordinary numbers. A scalar is described by magnitude only, but a vector has both magnitude and direction. Because of that, physicists use special symbols so the reader can immediately see whether a quantity is a vector or not.

A vector is often written in boldface, such as $\mathbf{A}$, or with an arrow above it, such as $\vec{A}$. Both mean the same thing. A scalar is written in ordinary type, such as $A$ or $t$ or $m$. This difference is very important, because $\mathbf{A}$ and $A$ do not mean the same thing. The symbol $\mathbf{A}$ refers to the full vector, while $A$ often refers to its magnitude.

A vector and its magnitude are different quantities.
If $\mathbf{A}$ is a vector, then its magnitude is written as
$$A = |\mathbf{A}|$$
Do not confuse $\mathbf{A}$ with $A$.

Common Ways to Write Vectors

In beginner physics, you will usually see vectors written in one of these forms.

MeaningCommon notation
Vector A$\vec{A}$ or $\mathbf{A}$
Magnitude of A$\vec{A}$ or $A$
Vector from point P to point Q$\overrightarrow{PQ}$
Zero vector$\vec{0}$

The notation $\overrightarrow{PQ}$ means a vector that starts at point $P$ and points toward point $Q$. It is not just the two points themselves. It represents both a length and a direction.

Magnitude Notation

The size of a vector is called its magnitude, or length. Magnitude is always a scalar. If a vector is written as $\vec{v}$, then its magnitude is written as $|\vec{v}|$.

For example, if a car moves east with velocity $\vec{v}$, then $|\vec{v}|$ tells you how fast it is moving, without mentioning the direction.

Magnitude is never negative.
For any vector $\vec{A}$,
$$|\vec{A}| \ge 0$$
Also,
$$|\vec{0}| = 0$$

Equality of Vectors

Two vectors are equal if they have the same magnitude and the same direction. Their position in space does not matter if we are talking about free vectors, which is the usual case in introductory physics.

So if one arrow is moved somewhere else on the page but still points the same way and has the same length, it represents the same vector.

Equal vectors at different positions

If $\vec{A} = \vec{B}$, then the two vectors have identical magnitude and direction.

Negative of a Vector

The negative of a vector has the same magnitude as the original vector, but points in the opposite direction. If a vector is $\vec{A}$, then its opposite is written as $-\vec{A}$.

A vector and its negative

This notation is useful in physics when describing opposite directions, such as right and left, upward and downward, or forces acting in opposite ways.

The vector $-\vec{A}$ is not the same as the magnitude $-A$.
Usually, $A = |\vec{A}|$ is nonnegative, while $-\vec{A}$ is a vector pointing in the opposite direction.

Naming Vectors in Physics

Many physical quantities are vectors, and notation helps identify them quickly. Common examples include displacement, velocity, acceleration, force, and electric field. You may see them written as $\vec{r}$, $\vec{v}$, $\vec{a}$, $\vec{F}$, and $\vec{E}$.

Physical quantityTypical vector symbolTypical scalar related to it
Displacement$\vec{r}$distance or $\vec{r}$
Velocity$\vec{v}$speed, $\vec{v}$
Acceleration$\vec{a}$$\vec{a}$
Force$\vec{F}$$\vec{F}$

The exact letter often gives a clue about the physical meaning, while the arrow or boldface tells you that direction matters.

Geometric Picture

A vector is often drawn as an arrow. The length of the arrow shows magnitude, and the arrowhead shows direction. This geometric picture is closely connected to the notation.

Vector as an arrow

When written on paper, the symbol $\vec{A}$ stands for this whole directed arrow, not just its length.

Zero Vector Notation

The zero vector is written as $\vec{0}$. It has magnitude zero and no specific direction. It is an important special case in vector notation because it behaves like zero in ordinary arithmetic, but it is still a vector.

For example, if an object’s displacement is $\vec{0}$, that means its final position is the same as its initial position.

The zero vector is written as
$$\vec{0}$$
It is a vector, not an ordinary scalar zero, even though its magnitude is zero.

Notation in Printed and Handwritten Work

In textbooks, vectors are often shown in boldface, like $\mathbf{v}$. In handwritten notes, boldface is hard to produce, so an arrow is usually placed above the symbol, like $\vec{v}$. You should be able to recognize both styles.

ContextCommon style
Printed textbook$\mathbf{A}$
Handwritten notes$\vec{A}$

It is best to stay consistent within one piece of work. If you begin by writing vectors with arrows, keep using arrows.

Reading Vector Expressions

A simple vector expression like
$$\vec{C} = \vec{A} + \vec{B}$$
is read as “vector C equals vector A plus vector B.” Even before learning how addition works, the notation already tells you that all three symbols are vectors.

Similarly,
$$|\vec{C}| = 5 \text{ m}$$
means the magnitude of vector $\vec{C}$ is 5 meters.

This kind of notation allows us to separate the full vector quantity from its size, which is one of the most useful ideas in physics writing.

Final Reminder

Vector notation is a visual signal. It tells you that direction matters. Whenever you see an arrow over a symbol or a bold symbol, you should think of a quantity with both magnitude and direction. Whenever you see vertical bars around a vector, you should think of magnitude only.

Key notation rules:
$$\text{Vector: } \vec{A} \text{ or } \mathbf{A}$$
$$\text{Magnitude: } |\vec{A}| \text{ or } A$$
$$\text{Opposite vector: } -\vec{A}$$
$$\text{Zero vector: } \vec{0}$$
Always distinguish clearly between a vector and its magnitude.

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