In AP Physics C Mechanics, unit vector notation makes multidimensional vectors concise and exact. It lets you represent direction algebraically and describe an object's location relative to the origin with standard symbols.

Unit Vector Notation

In three dimensions, a vector can be split into perpendicular components along the coordinate axes.

A right-handed $x$–$y$–$z$ coordinate system with the standard unit vectors $\hat{i}$, $\hat{j}$, and $\hat{k}$. This visual anchors unit vector notation by showing each unit vector as a direction-only arrow of length 1 along its axis. Source

Unit vector notation labels each component with a direction so the full vector can be written in one compact expression.

Because the axes are perpendicular, each component can be handled independently. The numbers multiplying the unit vectors are scalar components, and their signs show whether the vector points in the positive or negative axis direction.

This notation separates magnitude information from direction information. The component values tell you how much of the vector lies along each axis, while the unit vectors identify which axis each component belongs to.

A negative component does not mean a negative magnitude for the entire vector. It means that particular part of the vector points opposite the positive axis direction. For example, a negative $y$-component points in the $-\hat{j}$ direction.

Interpreting Components

When reading a vector written in unit vector form, focus on each component one axis at a time.

• $A_x$ measures the part of the vector along the $x$-axis.

• $A_y$ measures the part along the $y$-axis.

• $A_z$ measures the part along the $z$-axis.

• The full vector is the combination of all three perpendicular pieces.

• In two-dimensional situations, the $z$-component is often zero, so only $\hat{i}$ and $\hat{j}$ appear.

The unit vectors themselves do not change size. Each has magnitude 1. Their role is purely directional, which is why they are essential in mechanics notation: they let a vector be manipulated algebraically without losing information about direction.

Position Vectors

One especially important vector is the position vector. It specifies where an object is located relative to the origin of the chosen coordinate system.

For a position vector, the scalar components are the object's coordinates. That is why position written as a vector looks closely related to coordinate notation, but the arrow on the symbol shows that the quantity has both magnitude and direction.

A position vector always begins at the origin and ends at the object. This matters physically and mathematically. If the origin is changed, the position vector changes even when the object itself stays in the same place. The notation therefore always depends on the chosen coordinate system.

Using $r$ and $\hat{r}$

The same position vector can also be written in a radial form that separates its length from its direction.

A polar-coordinate diagram showing the position vector $\vec{r}$ from the origin to a point $P$ and the corresponding unit radial direction $\hat{r}$. It makes the separation in $\vec{r}=r\hat{r}$ explicit: $r$ sets the distance, while $\hat{r}$ sets the direction. Source

This is the meaning of expressing position with $r$ and r-hat.

Here, $r$ is a scalar distance, while $\hat{r}$ gives direction only. The vector $\vec{r}$ and the unit vector $\hat{r}$ point the same way, but they do not have the same magnitude. The magnitude of $\hat{r}$ is always 1, while the magnitude of $\vec{r}$ is $r$.

This distinction is central to correct notation. Writing $\vec{r}$ means a full vector. Writing $r$ means only its size. Writing $\hat{r}$ means only its direction. On AP Physics C problems, mixing these symbols can turn a correct physical idea into an incorrect mathematical statement.

Relating Cartesian and Radial Descriptions

A single physical location can therefore be described in two consistent ways. In Cartesian form, the position is built from fixed axis directions $\hat{i}$, $\hat{j}$, and $\hat{k}$. In radial form, the same position is built from a distance $r$ and a direction $\hat{r}$.

These are not different positions. They are two notational descriptions of the same vector. A strong AP Physics C student should be able to recognize whether a problem is emphasizing fixed coordinate components or direction from the origin, and then interpret $\vec{r}$ appropriately.

Common Notation Points

• Do not confuse $\vec{r}$ with $r$.

• Do not attach units to $\hat{i}$, $\hat{j}$, $\hat{k}$, or $\hat{r}$.

• Coordinates such as $x$, $y$, and $z$ are scalars, not vectors by themselves.

• A missing hat changes a unit vector into something else entirely.

• A missing arrow changes a full vector into a scalar symbol or component.

• Careful notation is part of correct physics communication, not just formatting.