How Is Angular Displacement Related to the Displacement Vector?

Angular displacement and a displacement vector describe different aspects of motion. Angular displacement measures a change in orientation around an axis. A displacement vector records the change in position, pointing directly from the starting point to the ending point. They are related for rotational motion, but one cannot generally be substituted for the other.

What each quantity means

If an object moves from position vector $\mathbf{r}_i$ to position vector $\mathbf{r}_f$, its displacement is

$$\Delta\mathbf{r}=\mathbf{r}_f-\mathbf{r}_i.$$

This is a vector with units of length. It depends only on the initial and final positions, not on the route taken between them.

For motion in a plane, angular displacement is the signed change in angle:

$$\Delta\theta=\theta_f-\theta_i.$$

It is commonly measured in radians. Its sign indicates the chosen direction of rotation: for example, counterclockwise is often positive. Angular displacement is not the same as the distance traveled along the path. In three dimensions, rotation about a specified fixed axis can be represented by an angular-displacement vector whose direction is along that axis, but this does not make it the object’s ordinary position-displacement vector.

The relation for circular motion

Suppose a point moves along a circle of radius $R$ and turns through angle $\Delta\theta$. The signed arc length is

$$s=R\Delta\theta,$$

when the angle is in radians. The path length for a single movement without reversing is $R|\Delta\theta|$.

The displacement vector instead runs straight from the starting point to the ending point. That straight segment is a chord, so its magnitude is

$$|\Delta\mathbf{r}|=2R\left|\sin\left(\frac{\Delta\theta}{2}\right)\right|.$$

The arc and chord are close in length for a small angular change, but they are not equal in general. For a very small angle, $\sin(\Delta\theta/2)$ is approximately $\Delta\theta/2$, so the chord magnitude is approximately $R|\Delta\theta|$.

For a fixed rotation axis that passes through the origin and is represented by unit vector $\hat{\mathbf{n}}$, an infinitesimal position change can also be written

$$d\mathbf{r}=d\theta\,(\hat{\mathbf{n}}\times\mathbf{r}).$$

This gives the tangential direction of motion. For a finite turn, the simple cross-product expression is not generally an exact formula for the total displacement; use the difference between the final and initial position vectors instead.

Worked example

A point moves on a circle of radius $2$ m from angle $30^\circ$ to $90^\circ$, measured counterclockwise from the positive $x$-axis. Its angular displacement is

$$\Delta\theta=90^\circ-30^\circ=60^\circ=\frac{\pi}{3}\text{ rad}.$$

The initial and final position vectors are

$$\mathbf{r}_i=2(\cos 30^\circ,\sin 30^\circ)=(\sqrt{3},1)\text{ m},$$ $$\mathbf{r}_f=2(\cos 90^\circ,\sin 90^\circ)=(0,2)\text{ m}.$$

Subtract to get the displacement vector:

$$\Delta\mathbf{r}=\mathbf{r}_f-\mathbf{r}_i=(-\sqrt{3},1)\text{ m}.$$

Its magnitude is $\sqrt{3+1}=2$ m. The arc length, however, is

$$s=R\Delta\theta=2\left(\frac{\pi}{3}\right)=\frac{2\pi}{3}\text{ m}\approx2.09\text{ m}.$$

Thus the point travels about $2.09$ m along the circle, while its displacement magnitude is $2$ m. The displacement vector points from the initial location to the final location, not along the circular path.

What happens after a full turn?

If the point completes one full revolution and returns to its starting position, its angular displacement is $2\pi$ rad (with the sign set by direction), but its displacement vector is $\mathbf{0}$. This makes the distinction especially clear: angular displacement tracks rotation, whereas displacement tracks the net change in position.

For a full turn, the distance traveled along the circle is $2\pi R$. If the motion includes multiple turns, the signed angular displacement can be greater than $2\pi$ in magnitude even though the initial and final positions may coincide.

Common mistakes

• Equating arc length with displacement: Arc length measures distance along the curve; displacement is the straight-line vector between endpoints.
• Treating angular displacement as a length: Radians are dimensionless, while linear displacement has units such as metres. Multiplying the angle by the radius gives an arc length, not a displacement vector.
• Ignoring direction: The sign of angular displacement depends on the chosen positive rotation direction. A vector displacement has its own direction, determined by the endpoint positions.
• Assuming endpoint positions reveal the number of turns: The same final position can follow a short turn, a full revolution, or several revolutions. The displacement vector alone does not record that history.
• Calling every angular change a 3D vector: In planar problems it is standard to use a signed scalar angle. A vector representation in three dimensions requires a specified rotation axis; finite rotations about different axes also do not combine by ordinary vector addition.

Use angular displacement when you need to describe rotation, and use $\Delta\mathbf{r}$ when you need the net change in position. For circular motion, the radius and angular change let you calculate both the path length and the endpoint-to-endpoint displacement.