Vector Applications to Kinematics

students, imagine tracking a drone, a football, or a car as it moves through space 🚁⚽🚗. Kinematics is the study of motion without first asking what causes it. In this lesson, you will use vectors to describe position, displacement, velocity, and acceleration in one, two, and three dimensions. This is powerful because vectors give both size and direction, which makes them ideal for modeling real motion.

By the end of this lesson, you should be able to:

explain the key terms in vector kinematics,
use vector equations to model position and motion,
find velocity and acceleration from position vectors,
connect vector ideas to geometry and trigonometry,
solve IB-style problems involving motion in a straight line, a plane, or space.

Vectors appear all over the Geometry and Trigonometry course because they link algebra, geometry, and physical movement. They help describe lines, distances, directions, and angles, which is why they are so useful for motion problems.

1. Describing Motion with Position Vectors

A particle’s position at time $t$ is often written as a vector. In two dimensions, a position vector can be written as $\mathbf{r}(t)=\begin{pmatrix}x(t)\y(t)\end{pmatrix}$, and in three dimensions as $\mathbf{r}(t)=\begin{pmatrix}x(t)\y(t)\z(t)\end{pmatrix}$. Each component tells you where the particle is along that axis.

For example, if a drone moves so that

$$\mathbf{r}(t)=\begin{pmatrix}2t+1\t^2\end{pmatrix},$$

then at time $t=0$, its position is $\begin{pmatrix}1\\0\end{pmatrix}$, and at time $t=3$, its position is $\begin{pmatrix}7\\9\end{pmatrix}$. The vector shows the full position, not just a single number.

The displacement from time $t_1$ to time $t_2$ is the change in position:

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

This is different from total distance travelled. Displacement is a vector, so it has direction. Distance is a scalar, so it only measures how much ground was covered. A person walking around a block might travel a long distance but end up close to where they started.

2. Velocity and Acceleration as Vector Derivatives

In kinematics, velocity is the rate of change of position with respect to time. If position is given by $\mathbf{r}(t)$, then velocity is

$$\mathbf{v}(t)=\frac{d\mathbf{r}}{dt}.$$

Acceleration is the rate of change of velocity:

$$\mathbf{a}(t)=\frac{d\mathbf{v}}{dt}=\frac{d^2\mathbf{r}}{dt^2}.$$

Because differentiation acts on each component separately, if

$$\mathbf{r}(t)=\begin{pmatrix}x(t)\y(t)\z(t)\end{pmatrix},$$

then

$$\mathbf{v}(t)=\begin{pmatrix}x'(t)\y'(t)\z'(t)\end{pmatrix} \quad \text{and} \quad \mathbf{a}(t)=\begin{pmatrix}x''(t)\y''(t)\z''(t)\end{pmatrix}.$$

Example: suppose

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

Then

$$\mathbf{v}(t)=\begin{pmatrix}2t\\3\end{pmatrix}, \quad \mathbf{a}(t)=\begin{pmatrix}2\\0\end{pmatrix}.$$

At $t=2$, the velocity is $\begin{pmatrix}4\\3\end{pmatrix}$. This means the particle is moving 4 units per second in the $x$-direction and 3 units per second in the $y$-direction at that instant.

A very important idea is that velocity and acceleration are vectors, so direction matters. A particle can speed up, slow down, or change direction. For example, if velocity and acceleration point in the same direction, the speed usually increases. If they point in opposite directions, the speed usually decreases.

3. Straight-Line Motion and Constant Acceleration

Many IB questions involve motion in a straight line or motion that can be separated into components. If a particle moves with constant acceleration, then each coordinate follows a quadratic model. This is closely related to the standard equations of motion from physics.

For one-dimensional motion, the formulas are:

$$v=u+at,$$

$$s=ut+\tfrac12 at^2,$$

$$v^2=u^2+2as,$$

where $u$ is initial velocity, $v$ is velocity at time $t$, $a$ is acceleration, and $s$ is displacement.

In vector form, a similar idea applies to each component. If acceleration is constant, then position is a vector quadratic function of time.

Example: a particle moves in the plane with

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

Its velocity is

$$\mathbf{v}(t)=\begin{pmatrix}2t+2\\4\end{pmatrix}.$$

Its acceleration is

$$\mathbf{a}(t)=\begin{pmatrix}2\\0\end{pmatrix}.$$

Here the acceleration is constant, so the motion in the $x$-direction is speeding up regularly, while the $y$-motion has constant velocity.

If you are asked when the particle is at rest, you solve $\mathbf{v}(t)=\mathbf{0}$. That means every component must equal zero at the same time. If only one component is zero, the particle may still be moving in another direction.

4. Speed, Direction, and the Link to Trigonometry

The speed of a particle is the magnitude of its velocity:

$$|\mathbf{v}(t)|.$$

$$\mathbf{v}(t)=\begin{pmatrix}a\b\end{pmatrix},$$

then its speed is

$$|\mathbf{v}(t)|=\sqrt{a^2+b^2}.$$

In three dimensions,

$$|\mathbf{v}(t)|=\sqrt{a^2+b^2+c^2}.$$

This is where trigonometry and geometry connect strongly. The magnitude formula comes from Pythagoras’ theorem, which is a geometric result. If the velocity vector makes an angle $\theta$ with the positive $x$-axis, then

$$\cos \theta=\frac{a}{|\mathbf{v}|} \quad \text{and} \quad \sin \theta=\frac{b}{|\mathbf{v}|}.$$

This helps you find direction from components.

Example: if the velocity is

$$\mathbf{v}=\begin{pmatrix}3\\4\end{pmatrix},$$

then the speed is

$$|\mathbf{v}|=5.$$

The direction angle satisfies

$$\tan \theta=\frac{4}{3},$$

so the motion is about $53.1^\circ$ above the positive $x$-axis.

This kind of reasoning appears often in IB questions because it combines vector algebra with trigonometry in a meaningful way 📐.

5. Relative Motion and Meeting Problems

Vector kinematics is especially useful when two moving objects interact. You may be asked whether two particles meet, when they are closest, or whether one is catching another.

If particle $A$ has position vector $\mathbf{r}_A(t)$ and particle $B$ has position vector $\mathbf{r}_B(t)$, then they meet when

$$\mathbf{r}_A(t)=\mathbf{r}_B(t).$$

This means their coordinates are equal at the same time.

Example: suppose

$$\mathbf{r}_A(t)=\begin{pmatrix}2t\t+1\end{pmatrix} \quad \text{and} \quad \mathbf{r}_B(t)=\begin{pmatrix}t+3\\3-t\end{pmatrix}.$$

To find a meeting time, solve the system

$$2t=t+3$$

and

$$t+1=3-t.$$

Both give $t=3$, so the particles meet at $t=3$.

Sometimes the question asks for the distance between particles. If the position vectors are $\mathbf{r}_A(t)$ and $\mathbf{r}_B(t)$, the separation vector is

$$\mathbf{r}_B(t)-\mathbf{r}_A(t).$$

Its magnitude is the distance between them:

$$\left|\mathbf{r}_B(t)-\mathbf{r}_A(t)\right|.$$

To find the time when they are closest, you often minimize the square of the distance because it is easier to differentiate:

$$\left|\mathbf{r}_B(t)-\mathbf{r}_A(t)\right|^2.$$

This is a standard HL technique because it combines algebra, calculus, and vectors.

6. How Vector Kinematics Fits Geometry and Trigonometry

Vector applications to kinematics are not a separate island; they sit right inside Geometry and Trigonometry. Here is why:

Vectors describe line directions and plane movement.
Magnitudes use distance formulas from geometry.
Angles between vectors use trigonometric ideas.
Motion questions often involve coordinates, gradients, and intersections.

For example, the angle between two vectors $\mathbf{a}$ and $\mathbf{b}$ can be found using the dot product:

$$\mathbf{a}\cdot\mathbf{b}=|\mathbf{a}||\mathbf{b}|\cos \theta.$$

This matters in kinematics when you want to know whether velocity and acceleration are aligned, perpendicular, or pointing in opposite directions. If $\mathbf{v}\cdot\mathbf{a}=0$, then the vectors are perpendicular. In motion, that can mean the particle’s speed is not increasing in the direction of motion at that instant.

A 3D motion example could describe a satellite or aircraft. If

$$\mathbf{r}(t)=\begin{pmatrix}t\\2t\\5-t\end{pmatrix},$$

then

$$\mathbf{v}(t)=\begin{pmatrix}1\\2\\-1\end{pmatrix},$$

which is constant. That means the object moves in a straight line with constant speed. Such models help you connect geometry in space with the real-world path of an object.

Conclusion

students, vector kinematics gives you a clear mathematical language for motion. Position vectors tell you where something is, displacement tells you how far and in what direction it moved, velocity tells you how position changes over time, and acceleration tells you how velocity changes. By using differentiation, magnitudes, and the dot product, you can solve motion problems in $2$D and $3$D with accuracy.

This topic is important because it links many parts of IB Mathematics: Analysis and Approaches HL. It uses algebraic manipulation, calculus, trigonometry, and geometry together. When you understand vector applications to kinematics, you are better prepared for questions about moving particles, relative motion, and motion in space.

Study Notes

A position vector is written as $\mathbf{r}(t)$ and gives the location of a particle at time $t$.
Displacement is $\Delta \mathbf{r}=\mathbf{r}(t_2)-\mathbf{r}(t_1)$.
Velocity is the derivative of position: $\mathbf{v}(t)=\frac{d\mathbf{r}}{dt}$.
Acceleration is the derivative of velocity: $\mathbf{a}(t)=\frac{d\mathbf{v}}{dt}=\frac{d^2\mathbf{r}}{dt^2}$.
Speed is the magnitude of velocity: $|\mathbf{v}(t)|$.
For two particles to meet, set their position vectors equal: $\mathbf{r}_A(t)=\mathbf{r}_B(t)$.
The distance between two particles is the magnitude of the separation vector: $\left|\mathbf{r}_B(t)-\mathbf{r}_A(t)\right|$.
The dot product formula is $\mathbf{a}\cdot\mathbf{b}=|\mathbf{a}||\mathbf{b}|\cos \theta$.
Vector kinematics connects directly to geometry, trigonometry, and calculus.
In IB HL problems, always check units, time domain, and whether a vector equation is in $2$D or $3$D.