Vectors
Introduction
students, imagine a drone flying across a city 🌆. Its movement is not just about how far it goes, but also which direction it travels. That is the idea behind a vector: a quantity that has both magnitude and direction. In IB Mathematics: Applications and Interpretation HL, vectors help you describe position, movement, shape, and relationships in two and three dimensions.
In this lesson, you will learn how vectors are written, how to find their length and direction, and how to use them to solve geometry problems. You will also see how vectors connect to other areas of Geometry and Trigonometry, such as coordinates, lines, and angles. By the end, you should be able to explain the main ideas and terminology behind vectors, apply common vector methods, and recognize why vectors are useful in real-world spatial reasoning.
What Is a Vector?
A scalar is a quantity with size only, such as $5\text{ m}$, $12\text{ s}$, or $30\degree$. A vector has both size and direction. Examples include velocity, force, and displacement. If a car moves $20\text{ m}$ east, the distance is a scalar, but the displacement is a vector because east is part of the description 🚗.
Vectors are often shown in several ways:
- as a directed line segment with an arrow
- in bold type, such as $\mathbf{v}$
- using column form, such as $\begin{pmatrix}2\\-1\end{pmatrix}$ in 2D or $\begin{pmatrix}1\\3\\-2\end{pmatrix}$ in 3D
- using unit vectors, such as $\mathbf{i}$, $\mathbf{j}$, and $\mathbf{k}$
A position vector describes where a point is relative to the origin. If $P$ has coordinates $(x,y)$, then its position vector is $\begin{pmatrix}x\y\end{pmatrix}$. In three dimensions, if $P$ has coordinates $(x,y,z)$, then its position vector is $\begin{pmatrix}x\y\z\end{pmatrix}$.
The vector from point $A$ to point $B$ is written $\overrightarrow{AB}$. If $A=(x_1,y_1)$ and $B=(x_2,y_2)$, then
$$\overrightarrow{AB}=\begin{pmatrix}x_2-x_1\y_2-y_1\end{pmatrix}$$
This idea is central in coordinate geometry because it lets you measure movement between points without drawing a scale diagram every time.
Vector Operations and Meaning
Vectors can be added, subtracted, and multiplied by scalars.
If $\mathbf{u}=\begin{pmatrix}a\b\end{pmatrix}$ and $\mathbf{v}=\begin{pmatrix}c\d\end{pmatrix}$, then
$$\mathbf{u}+\mathbf{v}=\begin{pmatrix}a+c\b+d\end{pmatrix}$$
$$\mathbf{u}-\mathbf{v}=\begin{pmatrix}a-c\b-d\end{pmatrix}$$
$$k\mathbf{u}=\begin{pmatrix}ka\kb\end{pmatrix}$$
These rules work the same way in three dimensions.
A helpful interpretation is that vector addition represents moving in stages. If you walk $3$ blocks north and then $4$ blocks east, the total displacement is the sum of those two vectors. This is useful in navigation, physics, and computer graphics 🎮.
Scalar multiplication changes the length of a vector. For example, $2\mathbf{u}$ points in the same direction as $\mathbf{u}$ but is twice as long, while $-\mathbf{u}$ points in the opposite direction.
A key IB idea is using vector equations to express relationships. For example, if $\mathbf{r}$ is the position vector of a moving point, then $\mathbf{r}=\mathbf{a}+\lambda\mathbf{d}$ describes a line, where $\mathbf{a}$ is a fixed point on the line, $\mathbf{d}$ is a direction vector, and $\lambda$ is a parameter. This is a powerful way to model lines in space.
Magnitude, Unit Vectors, and Direction
The magnitude of a vector is its length. If $\mathbf{v}=\begin{pmatrix}a\b\end{pmatrix}$, then
$$|\mathbf{v}|=\sqrt{a^2+b^2}$$
In three dimensions, if $\mathbf{v}=\begin{pmatrix}a\b\c\end{pmatrix}$, then
$$|\mathbf{v}|=\sqrt{a^2+b^2+c^2}$$
This formula comes from the Pythagorean theorem, which is why vectors fit naturally into Geometry and Trigonometry.
A unit vector has magnitude $1$. To find a unit vector in the direction of $\mathbf{v}$, divide by its magnitude:
$$\hat{\mathbf{v}}=\frac{\mathbf{v}}{|\mathbf{v}|}$$
If $\mathbf{v}=\begin{pmatrix}6\\8\end{pmatrix}$, then $|\mathbf{v}|=10$ and
$$\hat{\mathbf{v}}=\begin{pmatrix}\frac{6}{10}\\frac{8}{10}\end{pmatrix}=\begin{pmatrix}\frac{3}{5}\\frac{4}{5}\end{pmatrix}$$
This means the direction is the same, but the length is scaled to $1$.
Direction is often described using a direction vector or by comparing components. For example, the vector $\begin{pmatrix}2\\-3\end{pmatrix}$ goes $2$ units right and $3$ units down. In a map or CAD design, these components help describe exact movement 📐.
Using Vectors in Geometry
Vectors are excellent for proving geometric facts. One common use is showing that two vectors are parallel. If one vector is a scalar multiple of another, then they are parallel. For example, $\begin{pmatrix}4\\6\end{pmatrix}$ and $\begin{pmatrix}2\\3\end{pmatrix}$ are parallel because
$$\begin{pmatrix}4\\6\end{pmatrix}=2\begin{pmatrix}2\\3\end{pmatrix}$$
Vectors also help with collinearity. Three points $A$, $B$, and $C$ are collinear if the vectors $\overrightarrow{AB}$ and $\overrightarrow{AC}$ are scalar multiples of each other.
A midpoint can be found using vectors too. If $A$ and $B$ are points with position vectors $\mathbf{a}$ and $\mathbf{b}$, then the midpoint $M$ has position vector
$$\mathbf{m}=\frac{\mathbf{a}+\mathbf{b}}{2}$$
This is a compact way to solve coordinate problems.
Example: Let $A=(1,2)$ and $B=(5,8)$. Then the midpoint is
$$M=\left(\frac{1+5}{2},\frac{2+8}{2}\right)=(3,5)$$
Vectors also support geometric reasoning about triangles and parallelograms. If $\overrightarrow{AB}=\mathbf{u}$ and $\overrightarrow{AC}=\mathbf{v}$, then the fourth vertex of a parallelogram can often be found by vector addition. Since opposite sides are equal and parallel, vectors make the structure visible in algebraic form.
Vectors and Lines in 2D and 3D
In IB AI HL, lines are often written in vector form because this makes the direction and location of a line clear. A line through point with position vector $\mathbf{a}$ and direction vector $\mathbf{d}$ can be written as
$$\mathbf{r}=\mathbf{a}+\lambda\mathbf{d}$$
where $\lambda$ is a real number.
If the line passes through $(1,2)$ with direction vector $\begin{pmatrix}3\\-1\end{pmatrix}$, then
$$\mathbf{r}=\begin{pmatrix}1\\2\end{pmatrix}+\lambda\begin{pmatrix}3\\-1\end{pmatrix}$$
From this, the parametric equations are
$$x=1+3\lambda$$
$$y=2-\lambda$$
In three dimensions, this idea becomes even more useful. For example, a line in space may be written as
$$\mathbf{r}=\begin{pmatrix}2\\-1\\4\end{pmatrix}+\lambda\begin{pmatrix}1\\3\\2\end{pmatrix}$$
This means the line passes through $(2,-1,4)$ and moves in the direction $(1,3,2)$.
A vector approach is especially effective when determining whether lines are parallel or intersect. If two lines have direction vectors that are scalar multiples, they are parallel. If they can be written with the same point for some values of parameters, they intersect.
Real-World Applications
Vectors are not just abstract symbols. Engineers use them to model forces on bridges, architects use them to describe structure and symmetry, and pilots use them to account for wind and direction. A plane moving at $400\text{ km/h}$ north with wind pushing it east at $50\text{ km/h}$ experiences two velocity vectors. The actual motion is found by combining them.
In computer animation, vectors control movement across the screen. In robotics, they help machines move from one location to another precisely. In physics, force vectors can be added to find the resultant force, which determines how an object moves ⚙️.
These examples show why vectors are important in spatial reasoning. They turn movement and geometry into algebra, making complex situations easier to analyze.
Conclusion
students, vectors are a core tool in Geometry and Trigonometry because they describe both size and direction. They help you model positions, movements, lines, midpoints, parallelism, and three-dimensional space. By learning vector notation, magnitude, unit vectors, and vector equations of lines, you gain a flexible method for solving problems that appear in mathematics and in real-world contexts. Vectors connect algebra, geometry, and trigonometry into one powerful language of space.
Study Notes
- A vector has both magnitude and direction.
- A scalar has magnitude only.
- A position vector shows the location of a point relative to the origin.
- The vector from $A$ to $B$ is found by subtracting coordinates.
- If $\mathbf{u}$ and $\mathbf{v}$ are vectors, then $\mathbf{u}+\mathbf{v}$ and $k\mathbf{u}$ are found component by component.
- 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 unit vector is found by dividing a vector by its magnitude.
- Two vectors are parallel if one is a scalar multiple of the other.
- The midpoint of points with position vectors $\mathbf{a}$ and $\mathbf{b}$ is $\frac{\mathbf{a}+\mathbf{b}}{2}$.
- A line can be written as $\mathbf{r}=\mathbf{a}+\lambda\mathbf{d}$.
- Vectors are widely used in geometry, navigation, physics, engineering, and computer graphics.