Welcome to the World of Vectors!
In this chapter of Pure Mathematics, we are going to explore Vectors. Don’t worry if the name sounds a bit like something from a sci-fi movie; vectors are actually very simple tools that help us describe movement and position. While a normal number (a scalar) just tells us "how much" (like 5 kg or 10 minutes), a vector tells us two things: magnitude (size) and direction.
Think of it like this: If I tell you "The treasure is 10 meters away," you don't know where to dig. But if I say "The treasure is 10 meters North," you have a vector!
1. Understanding Vectors in 2D
In your AS Level course, we focus on vectors in two dimensions. We usually write them in two ways:
Column Vectors
A column vector looks like this: \( \binom{x}{y} \).
The top number \( x \) tells you how far to move horizontally (right is positive, left is negative).
The bottom number \( y \) tells you how far to move vertically (up is positive, down is negative).
Unit Vector Notation (\(\mathbf{i}\) and \(\mathbf{j}\))
We use two special "building block" vectors:
\(\mathbf{i}\) is a vector of length 1 going in the positive \( x \) direction.
\(\mathbf{j}\) is a vector of length 1 going in the positive \( y \) direction.
So, the vector \( 3\mathbf{i} + 4\mathbf{j} \) just means "3 steps right and 4 steps up."
Quick Review: \( \binom{3}{4} \) is the exact same thing as \( 3\mathbf{i} + 4\mathbf{j} \).
2. Magnitude and Direction
Sometimes we need to convert our "steps right and up" into a single distance and an angle.
Magnitude (How long is it?)
To find the magnitude (length) of a vector \( \mathbf{a} = x\mathbf{i} + y\mathbf{j} \), we use Pythagoras’ Theorem. We write magnitude using vertical bars: \( |\mathbf{a}| \).
\( |\mathbf{a}| = \sqrt{x^2 + y^2} \)
Direction (Which way is it pointing?)
We usually measure the direction as an angle \( \theta \) from the positive \( x \)-axis. We can find this using trigonometry:
\( \tan \theta = \frac{y}{x} \)
Common Mistake: When finding the angle, always draw a quick sketch! If your vector is \( -3\mathbf{i} - 4\mathbf{j} \), it's pointing into the bottom-left quadrant. Your calculator might give you a positive angle, but the sketch will show you that you need to add \( 180^\circ \) to get the correct direction.
Unit Vectors
A unit vector is simply any vector with a magnitude of 1. If you have a vector \( \mathbf{a} \) and you want to find a unit vector in that same direction, just divide the vector by its magnitude:
Unit vector \( \mathbf{\hat{a}} = \frac{\mathbf{a}}{|\mathbf{a}|} \)
Key Takeaway: Magnitude is the distance (Pythagoras), and Direction is the angle (Trigonometry).
3. Vector Arithmetic
Working with vectors is very similar to basic algebra. Don't let the bold letters scare you!
Addition and Subtraction
To add or subtract vectors, just do it component by component:
\( \binom{2}{3} + \binom{4}{-1} = \binom{2+4}{3-1} = \binom{6}{2} \)
Diagrammatically, adding vectors is like following a path. If you follow vector a and then follow vector b, the result is the vector a + b. This is called the Triangle Law.
Scalar Multiplication
You can multiply a vector by a normal number (a scalar). This "scales" the vector, making it longer or shorter, or reversing its direction if the number is negative.
Example: \( 3 \times \binom{2}{-5} = \binom{6}{-15} \)
Parallel Vectors
Two vectors are parallel if one is a scalar multiple of the other.
Example: \( \binom{1}{2} \) and \( \binom{3}{6} \) are parallel because \( \binom{3}{6} = 3 \times \binom{1}{2} \).
Did you know? If you multiply a vector by -1, it stays the same length but points in the exact opposite direction!
4. Position Vectors and Distance
A position vector is a vector that starts at the origin \( O(0,0) \). We usually write the position vector of point \( A \) as \( \vec{OA} \) or simply a.
The "AB" Rule
This is one of the most important tricks for your exam! If you want to find the vector that goes from point \( A \) to point \( B \), you use:
\( \vec{AB} = \mathbf{b} - \mathbf{a} \)
(Think: Finish minus Start).
Distance Between Two Points
To find the distance between two points \( (x_1, y_1) \) and \( (x_2, y_2) \), you are essentially finding the magnitude of the vector between them.
\( d^2 = (x_2 - x_1)^2 + (y_2 - y_1)^2 \)
Key Takeaway: To get from A to B, subtract the start position from the end position: \( \mathbf{b} - \mathbf{a} \).
5. Solving Geometric Problems
Vectors are great for proving things about shapes like parallelograms or triangles.
Ratios on a Line
Sometimes a point \( C \) splits a line \( AB \) into a ratio, like \( 1:2 \).
To find the position of \( C \):
1. Find the vector \( \vec{AB} \) (which is \( \mathbf{b} - \mathbf{a} \)).
2. Calculate the fraction of the way \( C \) is along the line (in a \( 1:2 \) ratio, \( C \) is \( \frac{1}{3} \) of the way).
3. Use the formula: \( \vec{OC} = \mathbf{a} + \frac{1}{3}(\mathbf{b} - \mathbf{a}) \).
Modeling with Vectors
In Pure Maths, you might see vectors used to describe forces or velocities.
- Resultant force is just the sum of all individual force vectors.
- If a particle is in equilibrium, the sum of all vectors is zero: \( \binom{0}{0} \).
Summary Tip: If a problem looks complicated, draw it! Most vector problems turn into simple triangles once you see them on paper.
Quick Review Checklist
● Can you switch between \( \binom{x}{y} \) and \( x\mathbf{i} + y\mathbf{j} \)?
● Do you remember that Magnitude is \( \sqrt{x^2 + y^2} \)?
● Do you know that \( \vec{AB} = \mathbf{b} - \mathbf{a} \)?
● Can you find a unit vector by dividing by the magnitude?
● Are you checking your angles with a sketch?
Don't worry if this seems tricky at first—vectors are a new way of thinking! Practice adding them and finding their lengths, and soon it will feel as natural as regular addition.