Position and Displacement Vectors

Welcome, students! In this lesson, you will learn how vectors describe location and movement in a clear and powerful way 🧭. Position and displacement vectors are essential tools in geometry, trigonometry, and later topics in physics and calculus. By the end of this lesson, you should be able to explain the key ideas, use vector notation correctly, and solve problems involving movement in $2D$ and $3D$ space.

Objectives:

Understand what a position vector is and how it describes a point.
Understand what a displacement vector is and how it describes movement from one point to another.
Use vector subtraction to find displacement.
Connect vectors to coordinate geometry, lines, and planes.
Solve IB-style problems using vector reasoning.

What is a Vector?

A vector is a quantity that has both magnitude and direction. Magnitude tells you “how much,” while direction tells you “which way.” For example, walking $5$ km north is a vector, because it includes a distance and a direction. Walking $5$ km alone would be only a scalar, because it gives size but no direction.

Vectors are written in different ways, such as column vectors, component form, or with an arrow above a letter. In coordinate geometry, a vector in two dimensions can be written as $\begin{pmatrix}x\y\end{pmatrix}$ and in three dimensions as $\begin{pmatrix}x\y\z\end{pmatrix}$.

The components show movement along each axis. For example, $\begin{pmatrix}3\\-2\end{pmatrix}$ means move $3$ units in the positive $x$-direction and $2$ units in the negative $y$-direction. This is a compact way to describe direction and distance together.

Position Vectors

A position vector describes the location of a point relative to a fixed origin, usually called $O$. If a point $P$ has coordinates $(x,y)$, then its position vector is $\overrightarrow{OP}=\begin{pmatrix}x\y\end{pmatrix}$ in two dimensions. In three dimensions, if $P$ has coordinates $(x,y,z)$, then $\overrightarrow{OP}=\begin{pmatrix}x\y\z\end{pmatrix}$.

This means a position vector is the vector from the origin to the point. It answers the question: “Where is the point?” This is very useful because a single vector can represent a point’s exact location in space.

For example, if $A=(4,-1)$, then the position vector of $A$ is $\overrightarrow{OA}=\begin{pmatrix}4\\-1\end{pmatrix}$.

If $B=(2,3,-5)$, then $\overrightarrow{OB}=\begin{pmatrix}2\\3\\-5\end{pmatrix}$.

A key idea is that the coordinate of a point and its position vector are closely connected. The point itself is not the vector, but its coordinates give the vector components from the origin. This becomes very useful when working with lines and shapes in coordinate geometry.

Displacement Vectors

A displacement vector describes the movement from one point to another. If point $A$ has position vector $\overrightarrow{OA}=\mathbf{a}$ and point $B$ has position vector $\overrightarrow{OB}=\mathbf{b}$, then the displacement from $A$ to $B$ is $\overrightarrow{AB}=\mathbf{b}-\mathbf{a}$.

This subtraction rule is one of the most important ideas in this lesson. It makes sense because to go from $A$ to $B$, you can think of going from the origin to $B$, then subtracting the journey to $A$.

For example, if $A=(1,2)$ and $B=(5,7),$ then

$$\overrightarrow{OA}=\begin{pmatrix}1\\2\end{pmatrix}, \qquad \overrightarrow{OB}=\begin{pmatrix}5\\7\end{pmatrix}$$

and so

$$\overrightarrow{AB}=\begin{pmatrix}5\\7\end{pmatrix}-\begin{pmatrix}1\\2\end{pmatrix}=\begin{pmatrix}4\\5\end{pmatrix}.$$

That means to move from $A$ to $B$, you travel $4$ units right and $5$ units up. The reverse displacement from $B$ to $A$ is

$$\overrightarrow{BA}=\begin{pmatrix}-4\\-5\end{pmatrix}.$$

Notice that displacement depends on direction. The distance between $A$ and $B$ is the length of the displacement vector, but the displacement vector itself also tells you direction.

Using Components and Magnitude

If a vector is $\mathbf{v}=\begin{pmatrix}x\y\end{pmatrix},$ its magnitude is written as $$|\mathbf{v}|=\sqrt{x^2+y^2}.$$

In three dimensions, if $\mathbf{v}=\begin{pmatrix}x\y\z\end{pmatrix},$ then $$|\mathbf{v}|=\sqrt{x^2+y^2+z^2}.$$

This formula comes from Pythagoras’ theorem. It measures the straight-line length of the vector.

Example: If $\overrightarrow{AB}=\begin{pmatrix}3\\4\end{pmatrix},$ then

$$|\overrightarrow{AB}|=\sqrt{3^2+4^2}=\sqrt{25}=5.$$

So the distance from $A$ to $B$ is $5$ units.

In real life, this could model a drone flying from one point to another. The displacement vector gives the route in terms of direction and movement, while the magnitude gives the actual straight-line distance. 🚁

Subtracting Position Vectors

A very common IB question asks you to find a displacement vector from two points. The method is always the same:

Write the position vectors of the two points.
Subtract the starting point vector from the ending point vector.
Simplify the result.

If $\overrightarrow{OA}=\begin{pmatrix}a_1\a_2\a_3\end{pmatrix}$ and $\overrightarrow{OB}=\begin{pmatrix}b_1\b_2\b_3\end{pmatrix},$ then

$$\overrightarrow{AB}=\begin{pmatrix}b_1-a_1\b_2-a_2\b_3-a_3\end{pmatrix}.$$

Example: Let $A=(-2,1,4)$ and $B=(3,-5,1)$. Then

$$\overrightarrow{OA}=\begin{pmatrix}-2\\1\\4\end{pmatrix}, \qquad \overrightarrow{OB}=\begin{pmatrix}3\\-5\\1\end{pmatrix}$$

$$\overrightarrow{AB}=\begin{pmatrix}3-(-2)\\-5-1\\1-4\end{pmatrix}=\begin{pmatrix}5\\-6\\-3\end{pmatrix}.$$

This tells you the exact movement from $A$ to $B$. If a coordinate is negative, that only means movement in the opposite direction of the positive axis.

Position and Displacement in Geometry and Trigonometry

Position and displacement vectors are not isolated ideas. They connect directly to the wider study of geometry and trigonometry.

In geometry, vectors help describe line segments, shapes, and relationships between points. For example, if two vectors are equal, they have the same direction and magnitude, even if they start from different places. This is useful when proving that sides of a quadrilateral are parallel or equal.

Vectors also help describe the equation of a line. A line can be written using a position vector and a direction vector. If a point on the line has position vector $\mathbf{a}$ and the line has direction vector $\mathbf{d}$, then another point on the line may be written as

$$\mathbf{r}=\mathbf{a}+\lambda\mathbf{d},$$

where $\lambda$ is a scalar. This idea is built from the same logic as displacement vectors: you start at one point and move in a given direction.

In trigonometry, vector components can be linked to angles. If a vector has magnitude $|\mathbf{v}|$ and makes an angle $\theta$ with the positive $x$-axis, then its components can be written as

$$\mathbf{v}=\begin{pmatrix}|\mathbf{v}|\cos\theta\\|\mathbf{v}|\sin\theta\end{pmatrix}$$

in two dimensions. This shows how vectors and trigonometry work together to describe direction precisely.

Worked Example: A Journey on a Grid

Suppose a student walks from point $P(2,1)$ to point $Q(8,6)$ on a coordinate grid. The position vectors are

$$\overrightarrow{OP}=\begin{pmatrix}2\\1\end{pmatrix}, \qquad \overrightarrow{OQ}=\begin{pmatrix}8\\6\end{pmatrix}.$$

The displacement vector is

$$\overrightarrow{PQ}=\begin{pmatrix}8\\6\end{pmatrix}-\begin{pmatrix}2\\1\end{pmatrix}=\begin{pmatrix}6\\5\end{pmatrix}.$$

The magnitude of this displacement is

$$|\overrightarrow{PQ}|=\sqrt{6^2+5^2}=\sqrt{61}.$$

So the student’s movement from $P$ to $Q$ is described by a vector with components $6$ and $5$, and the straight-line distance is $\sqrt{61}$ units. This is a classic example of how vectors simplify movement problems. 📍

Conclusion

Position vectors tell you where points are relative to the origin, and displacement vectors tell you how to move from one point to another. The key formula is

$$\overrightarrow{AB}=\mathbf{b}-\mathbf{a}.$$

Once you understand this, many other topics become easier, including lines, planes, and proofs in coordinate geometry. students, this lesson gives you a foundation for more advanced vector reasoning in IB Mathematics: Analysis and Approaches HL. The main idea is simple but powerful: vectors are a precise language for location and movement.

Study Notes

A vector has both magnitude and direction.
A position vector gives the location of a point relative to the origin.
If $P=(x,y)$, then $$\overrightarrow{OP}=\begin{pmatrix}x\y\end{pmatrix}.$$
If $P=(x,y,z)$, then $$\overrightarrow{OP}=\begin{pmatrix}x\y\z\end{pmatrix}.$$
The displacement from $A$ to $B$ is $$\overrightarrow{AB}=\mathbf{b}-\mathbf{a}.$$
The reverse displacement is $$\overrightarrow{BA}=-\overrightarrow{AB}.$$
The magnitude of $\begin{pmatrix}x\y\end{pmatrix}$ is $$\sqrt{x^2+y^2}.$$
The magnitude of $\begin{pmatrix}x\y\z\end{pmatrix}$ is $$\sqrt{x^2+y^2+z^2}.$$
Vectors help describe lines using $$\mathbf{r}=\mathbf{a}+\lambda\mathbf{d}.$$
Vectors connect coordinate geometry and trigonometry through direction, distance, and angle.