Introduction to Vectors
Welcome, students! 🌟 In this lesson, you will begin one of the most important ideas in Geometry and Trigonometry: vectors. Vectors help us describe movement, direction, position, and relationships in both $2$D and $3$D space. They are used in physics, navigation, engineering, robotics, and computer graphics, so they are not just a school topic — they are a powerful real-world tool.
By the end of this lesson, you should be able to:
- explain the meaning of a vector and the language used to describe it,
- distinguish between scalars and vectors,
- represent vectors using notation, components, and diagrams,
- perform basic vector operations,
- connect vectors to line and plane geometry later in the course.
Vectors are a foundation for many IB Mathematics: Analysis and Approaches HL ideas. They help you describe shapes and motion in a precise way, which is why they sit naturally inside Geometry and Trigonometry.
What is a vector?
A vector is a quantity that has both magnitude and direction. Magnitude means size or length, and direction tells you where it points. For example, if students walks $3$ km north, that movement is a vector because it has a size of $3$ km and a direction of north. If students walks $3$ km east instead, the magnitude is the same, but the vector is different because the direction changed.
A scalar has magnitude only. Examples include mass, time, temperature, and speed. A vector includes direction as well. Examples include displacement, velocity, force, and acceleration.
In mathematics, vectors are often written in bold or with an arrow, such as $\mathbf{v}$ or $\vec{v}$. In school mathematics, the arrow notation is common when showing direction on a diagram. A vector can be shown as an arrow from one point to another. The arrow’s length represents magnitude, and the arrowhead shows direction.
A key idea is that vectors do not depend on where they are drawn, only on their length and direction. This means the vector from one place to another can be moved parallel to itself without changing the vector.
Representing vectors in $2$D and $3$D
Vectors can be represented using coordinates. In $2$D, a vector may be written as $\begin{pmatrix} a \\ b \end{pmatrix}$ where $a$ is the horizontal component and $b$ is the vertical component. This means the vector moves $a$ units left or right and $b$ units up or down.
For example, the vector $\begin{pmatrix} 4 \\ 3 \end{pmatrix}$ means move $4$ units in the positive $x$-direction and $3$ units in the positive $y$-direction. If the vector were $\begin{pmatrix} -2 \\ 5 \end{pmatrix}$, it would move $2$ units left and $5$ units up.
In $3$D, a vector is written as $\begin{pmatrix} a \\ b \\ c \end{pmatrix}$ and the third component gives movement in the $z$-direction. This is important in spatial geometry, where objects do not lie flat on a page.
The size of a vector is called its magnitude or modulus. For a $2$D vector $\mathbf{v} = \begin{pmatrix} a \\ b \end{pmatrix}$ the magnitude is
$$|\mathbf{v}| = \sqrt{a^2+b^2}.$$
For a $3$D vector $\mathbf{v} = \begin{pmatrix} a \\ b \\ c \end{pmatrix}$ the magnitude is
$$|\mathbf{v}| = \sqrt{a^2+b^2+c^2}.$$
Example: if $\mathbf{v} = \begin{pmatrix} 6 \\ 8 \end{pmatrix},$ then
$$|\mathbf{v}| = \sqrt{6^2+8^2} = \sqrt{36+64} = \sqrt{100} = 10.$$
So the vector has length $10$ units. This formula comes from the Pythagorean theorem, which is why vectors connect strongly to geometry.
Position vectors and direction vectors
A position vector gives the position of a point relative to the origin. If point $A$ has coordinates $(x,y)$, then its position vector is $\overrightarrow{OA} = \begin{pmatrix} x \\ y \end{pmatrix}.$ In $3$D, if $A=(x,y,z)$, then
$$\overrightarrow{OA} = \begin{pmatrix} x \\ y \\ z \end{pmatrix}.$$
A direction vector describes the direction of a line or movement. Any non-zero vector parallel to a line can be a direction vector for that line. For example, the vector $\begin{pmatrix} 2 \\ -1 \end{pmatrix}$ and the vector $\begin{pmatrix} -4 \\ 2 \end{pmatrix}$ point in opposite directions along the same line because one is a scalar multiple of the other.
This idea is useful because vectors let us describe a line without needing a picture. If a line passes through the point with position vector $\mathbf{a}$ and has direction vector $\mathbf{d},$ then any point on the line can be written as
$$\mathbf{r} = \mathbf{a} + t\mathbf{d},$$
where $t$ is a real number. This is called a vector equation of a line. Even though this idea is explored more later, students should notice now how vectors give a compact and exact way to model geometry.
Basic vector operations
The most common vector operations are addition, subtraction, and multiplication by a scalar.
Addition
If $\mathbf{u} = \begin{pmatrix} a \\ b \end{pmatrix}$ and $\mathbf{v} = \begin{pmatrix} c \\ d \end{pmatrix},$ then
$$\mathbf{u}+\mathbf{v}=\begin{pmatrix} a+c \\ b+d \end{pmatrix}.$$
This means add matching components.
Example:
$$\begin{pmatrix} 3 \\ -2 \end{pmatrix} + \begin{pmatrix} 5 \\ 4 \end{pmatrix} = \begin{pmatrix} 8 \\ 2 \end{pmatrix}.$$
Subtraction
Similarly,
$$\mathbf{u}-\mathbf{v}=\begin{pmatrix} a-c \\ b-d \end{pmatrix}.$$
Example:
$$\begin{pmatrix} 7 \\ 1 \end{pmatrix} - \begin{pmatrix} 2 \\ 6 \end{pmatrix} = \begin{pmatrix} 5 \\ -5 \end{pmatrix}.$$
Scalar multiplication
If $k$ is a number and $\mathbf{u} = \begin{pmatrix} a \\ b \end{pmatrix},$ then
$$k\mathbf{u} = \begin{pmatrix} ka \\ kb \end{pmatrix}.$$
This changes the size of the vector, and if $k$ is negative, it also reverses the direction.
Example:
$$-2\begin{pmatrix} 3 \\ -1 \end{pmatrix} = \begin{pmatrix} -6 \\ 2 \end{pmatrix}.$$
The vector is stretched by a factor of $2$ and flipped in direction.
These operations are the building blocks for solving more advanced geometry problems. They help you find missing points, test whether lines are parallel, and describe movement between points.
Real-world meaning and geometric connections
Vectors appear everywhere in real life. In navigation, an airplane’s displacement depends on both distance and direction. In physics, force is a vector because pushing a box east is different from pushing it north, even if the force size is the same. In sport, a soccer ball’s motion depends on speed and direction, so velocity is a vector.
In geometry, vectors help describe shapes with precision. If students knows the position vectors of two points, then the vector between them can be found by subtraction. If $A$ and $B$ have position vectors $\mathbf{a}$ and $\mathbf{b},$ then
$$\overrightarrow{AB} = \mathbf{b}-\mathbf{a}.$$
This is one of the most important vector results in the course.
For example, if $A=(1,2)$ and $B=(5,7),$ then
$$\overrightarrow{AB} = \begin{pmatrix} 5 \\ 7 \end{pmatrix} - \begin{pmatrix} 1 \\ 2 \end{pmatrix} = \begin{pmatrix} 4 \\ 5 \end{pmatrix}.$$
This means moving from $A$ to $B$ requires $4$ units right and $5$ units up.
Vectors also help determine whether points are collinear. If one vector is a scalar multiple of another, they are parallel. This means the points lie on the same straight line. That kind of reasoning is central to coordinate geometry in IB AA HL.
Why vectors matter in this topic
Introduction to vectors is not just a separate section. It is the language used in much of the rest of Geometry and Trigonometry. Later, students will use vectors to study lines, intersections, angles between lines, and planes in $3$D. Vectors also connect to trigonometry through angles and component relationships, especially when using the dot product later in the topic.
The main reason vectors are so useful is that they turn geometric problems into algebraic ones. Instead of drawing several diagrams and guessing, you can write the problem using symbols and calculate the answer. This is especially valuable in $3$D geometry, where pictures can be hard to interpret accurately.
A strong understanding of vectors also improves problem solving. When a question asks for a direction, distance, midpoint, or relationship between shapes, vectors often provide a clean method. The skill is not just calculation; it is learning how to model a spatial situation clearly.
Conclusion
Vectors are a fundamental idea in Geometry and Trigonometry. They describe movement and position using both magnitude and direction, making them different from scalars. students should now understand how to write vectors in coordinate form, find their magnitude, and perform basic operations such as addition, subtraction, and scalar multiplication. These ideas are the starting point for more advanced work with lines, planes, and spatial reasoning in IB Mathematics: Analysis and Approaches HL. 💡
Study Notes
- A vector has magnitude and direction.
- A scalar has magnitude only.
- Vectors can be written as arrows or in component form such as $\begin{pmatrix} a \\ b \end{pmatrix}$ or $\begin{pmatrix} a \\ b \\ c \end{pmatrix}$.
- The magnitude of $\begin{pmatrix} a \\ b \end{pmatrix}$ is $\sqrt{a^2+b^2}$, and the magnitude of $\begin{pmatrix} a \\ b \\ c \end{pmatrix}$ is $\sqrt{a^2+b^2+c^2}$.
- A position vector gives the location of a point from the origin.
- A direction vector shows the direction of a line or movement.
- To find the vector from $A$ to $B$, use $\overrightarrow{AB}=\mathbf{b}-\mathbf{a}$.
- Add and subtract vectors by combining matching components.
- Multiply a vector by a scalar to change its length, and possibly its direction if the scalar is negative.
- Vectors are essential for later topics such as lines, planes, angles, and spatial geometry.
- Real-world examples include displacement, velocity, force, and navigation.