How Do Vectors Model Kinematics Problems (Position and Velocity)?

Vectors model kinematics by representing an object’s position with a position vector and its velocity with a vector whose direction and magnitude describe motion. For constant velocity, use $\mathbf{r}(t)=\mathbf{a}+t\mathbf{v}$, where $t$ is time.

This is Mathematics: Analysis and Approaches HL, AHL 3.14: Vector equation of a line. It can be examined analytically on Paper 1 or with GDC support on Paper 2.

How the Model Works

The vector $\mathbf{v}$ determines both direction and distance travelled per unit time. Between times $t_1$ and $t_2$, displacement is

$$\mathbf{r}(t_2)-\mathbf{r}(t_1)=(t_2-t_1)\mathbf{v}.$$

This follows because subtracting the two position vectors cancels the initial position.

Worked Example

Suppose distance is measured in metres and time in seconds:

$$\mathbf{r}(t)=\begin{pmatrix}2\\-1\\4\end{pmatrix}+t\begin{pmatrix}3\\2\\-1\end{pmatrix}.$$

At $t=5$, the position is

$$\mathbf{r}(5)=\begin{pmatrix}17\\9\\-1\end{pmatrix}.$$

Therefore, the particle is at the point $(17,9,-1)$ after five seconds. The displacement from $t=1$ to $t=5$ is

$$\mathbf{r}(5)-\mathbf{r}(1)=4\mathbf{v}=\begin{pmatrix}12\\8\\-4\end{pmatrix}.$$

Its velocity is $\mathbf{v}=\begin{pmatrix}3\\2\\-1\end{pmatrix}$, and its speed is

$$\lvert\mathbf{v}\rvert=\sqrt{3^2+2^2+(-1)^2}=\sqrt{14}\,\mathrm{m\,s^{-1}}.$$

Speed is scalar, so it has no direction.

Units must be consistent: if position is in metres and time is in seconds, velocity is in metres per second. A negative component indicates motion in the negative direction; it does not mean negative speed.

Common Misconception

A common misconception is that the velocity vector gives the object’s position. It does not: velocity describes how position changes, while substituting a time into $\mathbf{r}(t)$ gives position. Another error is confusing displacement with distance: displacement is a vector, whereas distance is a non-negative scalar.

Exam Technique

In exams, define each vector, substitute time, and show the magnitude calculation for speed. If asked whether two particles collide, equate their position vectors at the same time and solve all components consistently. Equal positions at different times do not represent a collision.