Vectors
Welcome, students, to one of the most useful ideas in geometry and trigonometry: vectors 📍. Vectors help us describe movement, direction, position, and change in a precise way. In real life, they appear in navigation, physics, engineering, computer graphics, and even sports tracking. If a car drives $5\,\text{km}$ east, or a drone moves $3\,\text{m}$ upward and $4\,\text{m}$ north, vectors are the language that captures both the size of the movement and the direction.
What a Vector Is
A vector is a quantity that has both magnitude and direction. Magnitude means “how much,” and direction means “which way.” A scalar, by contrast, has only magnitude, such as temperature or mass.
A vector can be written in several ways. In textbooks, a vector is often written as $\mathbf{v}$ or $\vec{v}$. In component form, a two-dimensional vector may be written as $\begin{pmatrix}x\y\end{pmatrix}$, and a three-dimensional vector as $\begin{pmatrix}x\y\z\end{pmatrix}$. These components show how far the vector moves horizontally, vertically, and, in 3D, forward/backward.
For example, the vector $\begin{pmatrix}3\\4\end{pmatrix}$ means a movement of $3$ units in one direction and $4$ units in a perpendicular direction. Its magnitude is found using Pythagoras’ theorem:
$$
$\left|\mathbf{v}\right|=\sqrt{3^2+4^2}=5$
$$
So this vector has length $5$ units. This connects vectors directly to geometry, because geometry studies shape, distance, and position.
Vector Notation and Basic Ideas
Vectors are usually represented by arrows on a diagram. The arrow’s length shows magnitude, and the arrowhead shows direction. Two vectors are equal if they have the same magnitude and the same direction, even if they are drawn in different places.
This is important: vectors are not fixed to one location. A vector is free to move anywhere on a diagram as long as its length and direction stay the same. That means the vector $\begin{pmatrix}2\\-1\end{pmatrix}$ can be drawn starting at different points, but it still represents the same movement.
The zero vector, written as $\begin{pmatrix}0\\0\end{pmatrix}$ or $\mathbf{0}$, has no magnitude and no direction. It is the vector version of “no movement.”
A unit vector has magnitude $1$. Unit vectors are useful because they show direction only. In many coordinate systems, the standard unit vectors are $\mathbf{i}=\begin{pmatrix}1\\0\end{pmatrix}$ and $\mathbf{j}=\begin{pmatrix}0\\1\end{pmatrix}$ in two dimensions, and $\mathbf{k}=\begin{pmatrix}0\\0\\1\end{pmatrix}$ in three dimensions.
Vector Arithmetic: Adding, Subtracting, and Multiplying
Vectors can be added and subtracted component by component. If
$$
$\mathbf{a}$=$\begin{pmatrix}$a_1\a_2$\end{pmatrix}$, \quad $\mathbf{b}$=$\begin{pmatrix}$b_1\b_2$\end{pmatrix}$
$$
then
$$
$\mathbf{a}+\mathbf{b}=\begin{pmatrix}a_1+b_1\a_2+b_2\end{pmatrix}$
$$
and
$$
$\mathbf{a}-\mathbf{b}=\begin{pmatrix}a_1-b_1\a_2-b_2\end{pmatrix}$
$$
This is easy to picture as walking in steps. If students walks $2$ blocks east and $3$ blocks north, then later walks another $4$ blocks east and $1$ block north, the total movement is $6$ blocks east and $4$ blocks north. So the combined vector is $\begin{pmatrix}6\\4\end{pmatrix}$.
Vectors can also be multiplied by a scalar, which is a number. If $k$ is a scalar and $\mathbf{v}=\begin{pmatrix}x\y\end{pmatrix}$, then
$$
$ k\mathbf{v}=\begin{pmatrix}kx\ky\end{pmatrix}$
$$
This changes the magnitude of the vector. If $k$ is positive, the direction stays the same. If $k$ is negative, the vector points in the opposite direction. For example, $-2\begin{pmatrix}1\\3\end{pmatrix}=\begin{pmatrix}-2\\-6\end{pmatrix}$.
These operations are essential in IB Mathematics: Applications and Interpretation SL because they allow students to model movement and combine routes, displacements, and forces in a structured way.
Position Vectors and Midpoints
A position vector tells us the location of a point relative to the origin. If a point $P$ has coordinates $(x,y)$, then its position vector is $\begin{pmatrix}x\y\end{pmatrix}$. In three dimensions, a point $P(x,y,z)$ has position vector $\begin{pmatrix}x\y\z\end{pmatrix}$.
Position vectors help describe geometry clearly. For instance, if point $A$ has position vector $\begin{pmatrix}1\\2\end{pmatrix}$ and point $B$ has position vector $\begin{pmatrix}5\\6\end{pmatrix}$, then the vector from $A$ to $B$ is
$$
\overrightarrow{AB}=$\begin{pmatrix}5$\\$6\end{pmatrix}$-$\begin{pmatrix}1$\\$2\end{pmatrix}$=$\begin{pmatrix}4$\\$4\end{pmatrix}$
$$
This means moving $4$ units right and $4$ units up.
Vectors also make midpoint problems straightforward. The midpoint of a line segment joining $A(x_1,y_1)$ and $B(x_2,y_2)$ is
$$
$\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$
$$
This formula is useful in coordinate geometry because it combines algebra with spatial reasoning. For example, the midpoint of $(2,1)$ and $(8,5)$ is $(5,3)$.
Magnitude, Direction, and Unit Vectors
The magnitude of a vector $\mathbf{v}=\begin{pmatrix}x\y\end{pmatrix}$ is
$$
$\left|\mathbf{v}\right|=\sqrt{x^2+y^2}$
$$
In three dimensions, if $\mathbf{v}=\begin{pmatrix}x\y\z\end{pmatrix}$, then
$$
$\left|\mathbf{v}\right|=\sqrt{x^2+y^2+z^2}$
$$
This gives the length of the vector in space. A vector like $\begin{pmatrix}6\\8\end{pmatrix}$ has magnitude $\sqrt{6^2+8^2}=10$.
If a vector has magnitude $m$, then a unit vector in the same direction is found by dividing by $m$:
$$
$\text{unit vector}=\frac{\mathbf{v}}{\left|\mathbf{v}\right|}$
$$
For $\mathbf{v}=\begin{pmatrix}3\\4\end{pmatrix}$, the unit vector is
$$
$\frac{1}{5}$$\begin{pmatrix}3$\\$4\end{pmatrix}$=$\begin{pmatrix}$$\frac{3}{5}$\$\frac{4}{5}$$\end{pmatrix}$
$$
Unit vectors are very useful when a direction matters more than the actual size, such as describing the path of a plane or the slope of a ramp 🚗.
Vectors in Geometry and Trigonometry
Vectors connect strongly to geometry because they describe relationships between points, lines, and shapes. For example, if $\overrightarrow{AB}$ and $\overrightarrow{CD}$ are equal, then the two directed segments have the same length and direction. This idea helps identify parallel lines and congruent segments.
Vectors also support trigonometry. When a vector makes an angle $\theta$ with the positive $x$-axis, its components can be written using trigonometric functions. For a vector of magnitude $r$,
$$
$\mathbf{v}=\begin{pmatrix}r\cos\theta\r\sin\theta\end{pmatrix}$
$$
This formula shows how trig helps break a vector into horizontal and vertical parts. It is a strong example of how Geometry and Trigonometry work together in IB.
Another useful idea is finding the angle between vectors. If $\mathbf{a}$ and $\mathbf{b}$ are vectors, then the dot product is
$$
$\mathbf{a}\cdot\mathbf{b}=|\mathbf{a}||\mathbf{b}|\cos\theta$
$$
where $\theta$ is the angle between them. In component form,
$$
$\mathbf{a}\cdot\mathbf{b}=a_1b_1+a_2b_2$
$$
for two-dimensional vectors, and similarly in three dimensions. This formula is powerful because it helps determine whether two vectors are perpendicular. If $\mathbf{a}\cdot\mathbf{b}=0$, then the vectors are at right angles.
For example, $\begin{pmatrix}2\\1\end{pmatrix}\cdot\begin{pmatrix}1\\-2\end{pmatrix}=2(1)+1(-2)=0$, so the vectors are perpendicular. This is a direct link to geometry and angle reasoning.
Three-Dimensional Interpretation
Vectors become even more useful in three dimensions because they help describe space clearly. A point in 3D is written as $(x,y,z)$, and a vector can show movement in three separate directions. This is important for modeling buildings, flights, and navigation in real space.
Suppose a drone moves from $A(1,2,3)$ to $B(4,6,5)$. Then the displacement vector is
$$
\overrightarrow{AB}=$\begin{pmatrix}4$-1\\6-2\\5-$3\end{pmatrix}$=$\begin{pmatrix}3$\\4\\$2\end{pmatrix}$
$$
Its magnitude is
$$
$\sqrt{3^2+4^2+2^2}=\sqrt{29}$
$$
This tells us the straight-line distance between the two points. In three-dimensional geometry, vectors are one of the clearest ways to describe position and movement.
Why Vectors Matter in IB Mathematics: Applications and Interpretation SL
In this course, vectors are not just abstract symbols. They are tools for modeling applied situations. Students may use vectors to describe routes, motion, directions, and geometric relationships. Vectors also help explain structure in coordinate geometry and provide a bridge to later topics such as three-dimensional problem solving.
When solving vector problems, students should think carefully about what each component means. Ask: What is the direction? What is the size? Is this a position vector, a displacement vector, or a unit vector? Careful interpretation is important because the same numbers can mean different things depending on context.
A strong vector solution usually includes accurate notation, a clear diagram when helpful, and step-by-step reasoning. This matches the broader IB approach of combining algebraic skill with interpretation and communication.
Conclusion
Vectors are a central part of Geometry and Trigonometry because they connect distance, direction, shape, and angle in one framework. They let us describe movement exactly, compare positions, calculate distances, and analyze relationships in both $2$D and $3$D. For students, mastering vectors means gaining a powerful language for solving problems in mathematics and in real-world applications 🌍.
Study Notes
- A vector has both magnitude and direction.
- A scalar has magnitude only.
- Vectors can be written as $\mathbf{v}$, $\vec{v}$, or in component form such as $\begin{pmatrix}x\y\end{pmatrix}$.
- Equal vectors have the same magnitude and direction.
- The zero vector is $\mathbf{0}$ and represents no movement.
- Vector addition and subtraction are done component by component.
- Scalar multiplication changes the size of a vector; a negative scalar reverses direction.
- A position vector gives the location of a point relative to the origin.
- The vector from $A$ to $B$ is found by subtracting position vectors: $\overrightarrow{AB}=\mathbf{b}-\mathbf{a}$.
- 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 $\sqrt{x^2+y^2+z^2}$.
- A unit vector has magnitude $1$ and gives direction only.
- The dot product helps find angles and test for perpendicular vectors.
- Vectors are essential in $2$D and $3$D geometry, trigonometry, and applied modeling.