Welcome to the World of Vectors!

In this chapter, we are going to explore Vectors. While you’ve been using numbers (like age or temperature) your whole life, vectors add a new dimension—literally! We use vectors to describe things that have both a size and a specific direction, like the wind blowing northwest or a car driving at 60 mph towards London.

By the end of these notes, you’ll be able to represent vectors, calculate their size, and use them to solve puzzles in both pure math and real-world mechanics. Don't worry if it feels like a lot of new symbols at first; once you see the patterns, it’s just like following a map!

1. What exactly is a Vector?

In Mathematics, we distinguish between two types of measurements:

  • Scalars: These only have magnitude (size). Examples: mass, time, distance, speed.
  • Vectors: These have both magnitude AND direction. Examples: force, velocity, displacement, acceleration.

How we write them down

To make sure we don’t confuse vectors with regular numbers, we use special notation:

  • Bold lowercase letters: \(\mathbf{a}\) or \(\mathbf{v}\). (When writing by hand, we usually underline them: a).
  • Two capital letters: \(\vec{AB}\) represents the vector starting at point \(A\) and ending at point \(B\).
  • Component Form: Using unit vectors \(\mathbf{i}\) (one step right) and \(\mathbf{j}\) (one step up), such as \(x\mathbf{i} + y\mathbf{j}\).
  • Column Vectors: Written as \(\binom{x}{y}\), where \(x\) is the horizontal movement and \(y\) is the vertical movement.

Quick Tip: Think of a vector as a set of instructions. \(\binom{3}{-2}\) just means "Take 3 steps right and 2 steps down."

Key Takeaway: A vector tells you how far to go and in which direction.

2. Magnitude and Direction

Sometimes we need to know the total "length" of a vector or the specific angle it points toward.

Magnitude (The Modulus)

The magnitude of a vector \(\mathbf{a} = \binom{x}{y}\) is written as \(|\mathbf{a}|\). Since a vector forms a right-angled triangle on a graph, we simply use Pythagoras’ Theorem!

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

Direction

The direction is the angle (\(\theta\)) the vector makes with the positive x-axis (measured counter-clockwise). We use basic trigonometry to find it:

\(\tan(\theta) = \frac{y}{x}\), so \(\theta = \tan^{-1}\left(\frac{y}{x}\right)\)

Note: Always draw a quick sketch of your vector to check which "quadrant" it is in so your angle makes sense!

Did you know? In navigation, we usually measure angles from North, but in Pure Math vectors, we almost always start from the positive x-axis (East).

Key Takeaway: Magnitude is the length (Pythagoras), and Direction is the angle (Trigonometry).

3. Basic Operations: Adding and Scaling

Working with vectors is very logical. You just handle the horizontal (\(x\)) and vertical (\(y\)) parts separately.

Vector Addition

To add two vectors, just add their components:

If \(\mathbf{a} = \binom{2}{3}\) and \(\mathbf{b} = \binom{4}{-1}\), then \(\mathbf{a} + \mathbf{b} = \binom{2+4}{3+(-1)} = \binom{6}{2}\).

Geometrical interpretation: This is called the Triangle Law. If you place the "tail" of vector \(\mathbf{b}\) at the "tip" of vector \(\mathbf{a}\), the resultant vector (\(\mathbf{a} + \mathbf{b}\)) is the shortcut from the start of \(\mathbf{a}\) to the end of \(\mathbf{b}\).

Scalar Multiplication

Multiplying a vector by a regular number (a scalar) changes its length but keeps it pointing in the same (or exactly opposite) direction.

\(3 \times \binom{2}{-1} = \binom{6}{-3}\)

If two vectors are parallel, one will always be a scalar multiple of the other (e.g., \(\binom{1}{2}\) is parallel to \(\binom{10}{20}\)).

Key Takeaway: To add vectors, add the tops and add the bottoms. Parallel vectors are just "scaled" versions of each other.

4. Position and Displacement Vectors

This is where vectors help us with Coordinate Geometry.

Position Vectors

A position vector describes the location of a point relative to the origin \(O(0,0)\). We usually write the position vector of point \(A\) as \(\vec{OA}\) or \(\mathbf{a}\).

Displacement Vectors (The "Bridge" Formula)

If you want to find the vector that gets you from point \(A\) to point \(B\), you use this very important rule:

\(\vec{AB} = \mathbf{b} - \mathbf{a}\)

Analogy: To find the difference between two positions, subtract where you started from where you ended up.

Distance Between Two Points

To find the distance between two points \(A(a, b)\) and \(B(c, d)\), you find the magnitude of the displacement vector \(\vec{AB}\):

\(Distance = \sqrt{(c-a)^2 + (d-b)^2}\)

Memory Aid: "B take away A" gets you from A to B!

Key Takeaway: \(\vec{AB} = \vec{OB} - \vec{OA}\). Position vectors start from the origin.

5. Special Types of Vectors

  • Equal Vectors: Two vectors are equal only if they have the same magnitude AND the same direction.
  • Unit Vectors: A vector with a magnitude of exactly 1. We often find a unit vector in the direction of \(\mathbf{a}\) by dividing the vector by its own magnitude: \(\hat{\mathbf{a}} = \frac{\mathbf{a}}{|\mathbf{a}|}\).
  • Resultant Vector: The vector produced by adding two or more vectors together.

6. Common Mistakes to Avoid

Don't worry if this seems tricky at first—even top mathematicians make these slips!

  • Mixing up Scalars and Vectors: You cannot add a scalar to a vector (e.g., \(5 + \binom{2}{3}\) is impossible!).
  • Negative Signs: Be extra careful when subtracting column vectors with negative components. \(\binom{2}{-3} - \binom{-1}{4} = \binom{2 - (-1)}{-3 - 4} = \binom{3}{-7}\).
  • Angle Direction: Remember that \(\tan^{-1}\) on a calculator often gives the smallest angle. Always check your sketch to see if you need to add \(180^\circ\) or \(360^\circ\) to get the correct direction from the positive x-axis.
  • Notation: Ensure you distinguish between the point \((3, 4)\) and the vector \(\binom{3}{4}\).

7. Quick Review Checklist

Can you...

  • Explain the difference between a scalar and a vector?
  • Write a vector in \(\mathbf{i}, \mathbf{j}\) form and column form?
  • Calculate the magnitude (length) using Pythagoras?
  • Find the resultant vector by adding components?
  • Calculate the distance between two points using \(\vec{AB} = \mathbf{b} - \mathbf{a}\)?

You've got this! Practice drawing the vectors out on graph paper if you're stuck—seeing the "arrows" makes the math much clearer.