Welcome to the World of Vectors!

In your math journey so far, you have mostly dealt with numbers that tell you "how much" of something there is—like a temperature of \(25^{\circ}C\) or a mass of \(5kg\). But what if you need to know where something is going? That is where Vectors come in!

Vectors are incredibly important because they describe the real world. From how a plane flies through the wind to how characters move in a video game, vectors are the "instructions" that tell us both the size and the direction of a movement. Don't worry if this seems a bit abstract at first; once you see how they work on a grid, it will feel just like followng a map!

1. What Exactly is a Vector?

To understand vectors, we need to distinguish them from Scalars:

  • Scalar: A value that only has magnitude (size).
    Example: Speed (50 mph), Distance (10 km).
  • Vector: A value that has both magnitude AND direction.
    Example: Velocity (50 mph North), Displacement (10 km East).
Analogy: The Treasure Map

If a treasure map says "Walk 10 steps," you won't find the gold because you don't know which way to go (that's a scalar). If the map says "Walk 10 steps North," you have a vector!

Key Takeaway: A vector is simply a quantity that tells you "how far" and "which way."

2. Representing Vectors (The Notation)

In Pure Mathematics (XPM01), you will see vectors written in two main ways. It is important to be comfortable with both!

A. Column Vectors

A vector is often written as a pair of numbers in a bracket: \(\begin{pmatrix} x \\ y \end{pmatrix}\).
The top number (\(x\)) tells you how far to move right (positive) or left (negative).
The bottom number (\(y\)) tells you how far to move up (positive) or down (negative).

B. \(\mathbf{i}\) and \(\mathbf{j}\) Notation

Sometimes we use "unit vectors" called \(\mathbf{i}\) and \(\mathbf{j}\):

  • \(\mathbf{i}\) is a jump of 1 unit to the right.
  • \(\mathbf{j}\) is a jump of 1 unit up.

So, the vector \(\begin{pmatrix} 3 \\ -2 \end{pmatrix}\) can also be written as \(3\mathbf{i} - 2\mathbf{j}\).

Did you know? In textbooks, vectors are usually bold (like \(\mathbf{a}\)). When you write them by hand, you should underline them (like \(\underline{a}\)) to show they aren't just regular numbers.

3. Magnitude: How Long is the Vector?

The magnitude of a vector is just its length. We use vertical bars to represent this: \(|\mathbf{a}|\).

To find the length of a vector \(\begin{pmatrix} x \\ y \end{pmatrix}\), we use our old friend Pythagoras’ Theorem!

Formula: \(|\mathbf{a}| = \sqrt{x^2 + y^2}\)

Example:

Find the magnitude of \(\mathbf{v} = \begin{pmatrix} 3 \\ 4 \end{pmatrix}\).
\(|\mathbf{v}| = \sqrt{3^2 + 4^2}\)
\(|\mathbf{v}| = \sqrt{9 + 16} = \sqrt{25} = 5\).

Common Mistake: When calculating the magnitude of a vector with a negative number, like \(\begin{pmatrix} 3 \\ -4 \end{pmatrix}\), remember that \((-4)^2\) is positive \(16\). Length can never be negative!

Key Takeaway: Magnitude = Length. Just use \(a^2 + b^2 = c^2\)!

4. Vector Arithmetic (Adding and Scaling)

Working with vectors is actually simpler than regular algebra in many ways.

Addition and Subtraction

To add or subtract vectors, just do it component-wise (top with top, bottom with bottom).
If \(\mathbf{a} = \begin{pmatrix} 2 \\ 5 \end{pmatrix}\) and \(\mathbf{b} = \begin{pmatrix} 3 \\ -1 \end{pmatrix}\):
\(\mathbf{a} + \mathbf{b} = \begin{pmatrix} 2+3 \\ 5+(-1) \end{pmatrix} = \begin{pmatrix} 5 \\ 4 \end{pmatrix}\).

Scalar Multiplication

This is just making a vector longer or shorter. If you multiply a vector by \(2\), it stays in the same direction but becomes twice as long.
\(2 \times \begin{pmatrix} 4 \\ -3 \end{pmatrix} = \begin{pmatrix} 8 \\ -6 \end{pmatrix}\).

Encouragement: Think of adding vectors like "joining journeys." If you walk from A to B, then B to C, the result (\(\mathbf{a}+\mathbf{b}\)) is just the shortcut from A to C!

5. Position and Displacement Vectors

This is a key concept for exam questions. There are two "types" of vectors based on where they start:

  • Position Vector: This always starts from the Origin (0,0). It tells you where a point is. We usually call the position vector of point \(A\) as \(\vec{OA}\) or \(\mathbf{a}\).
  • Displacement Vector: This is the journey between two points, like \(A\) to \(B\). We write this as \(\vec{AB}\).
The Magic Formula for \(\vec{AB}\)

To find the vector journey from \(A\) to \(B\), you use their position vectors:
\(\vec{AB} = \mathbf{b} - \mathbf{a}\)

Memory Aid: To get to \(\vec{AB}\), it’s always Second minus First.

Example:

Point \(A\) is at \((2, 3)\) and Point \(B\) is at \((5, 1)\). Find vector \(\vec{AB}\).
\(\mathbf{a} = \begin{pmatrix} 2 \\ 3 \end{pmatrix}\), \(\mathbf{b} = \begin{pmatrix} 5 \\ 1 \end{pmatrix}\)
\(\vec{AB} = \begin{pmatrix} 5 \\ 1 \end{pmatrix} - \begin{pmatrix} 2 \\ 3 \end{pmatrix} = \begin{pmatrix} 3 \\ -2 \end{pmatrix}\).

Key Takeaway: Use \(\mathbf{b} - \mathbf{a}\) to find the vector between two points.

6. Parallel Vectors

How do you know if two vectors are pointing in the same direction?
Two vectors are parallel if one is a multiple of the other.

Example: \(\begin{pmatrix} 2 \\ 3 \end{pmatrix}\) and \(\begin{pmatrix} 4 \\ 6 \end{pmatrix}\) are parallel because \(\begin{pmatrix} 4 \\ 6 \end{pmatrix} = 2 \times \begin{pmatrix} 2 \\ 3 \end{pmatrix}\).

Quick Review Box:
- Vector: Size and Direction.
- Magnitude: Use \(\sqrt{x^2 + y^2}\).
- \(\vec{AB}\): Calculate using \(\mathbf{b} - \mathbf{a}\).
- Parallel: One is a multiple of the other.

Final Summary

Vectors might feel like a new language, but they are just a way of describing movement on a grid. Remember that the top number is your horizontal movement and the bottom is your vertical movement. Most problems just require you to add, subtract, or use Pythagoras! Keep practicing drawing them out, and you'll master this chapter in no time.