Welcome to the World of Vectors!

Hello! Today we are diving into Vectors. If you’ve ever followed a map or played a video game where a character moves in a specific direction, you’ve already used vectors without knowing it. While a regular number (a scalar) just tells us "how much," a vector tells us "how much" and "which way."

In your AQA Paper 2 exam, you'll need to work with these in both 2D and 3D. Don't worry if it seems a bit abstract at first—we will break it down step-by-step!

1. What exactly is a Vector?

A scalar is just a size (magnitude), like your age or the temperature.
A vector has both magnitude (size) and direction.

How we write them

You will see vectors written in three main ways:

  • Bold letters: \(\mathbf{a}\) (usually in textbooks).
  • Underlined letters: \(\underline{a}\) (how you should write them in your exam!).
  • Component Form: Using unit vectors \(\mathbf{i}\), \(\mathbf{j}\), and \(\mathbf{k}\). These are just directions: \(\mathbf{i}\) is right, \(\mathbf{j}\) is up, and \(\mathbf{k}\) is "out" towards you in 3D.
  • Column Vectors: \(\begin{pmatrix} x \\ y \\ z \end{pmatrix}\). This is often the easiest way to do calculations!

Quick Review: In 3D, the vector \(3\mathbf{i} - 2\mathbf{j} + 5\mathbf{k}\) is the same as saying "Go 3 steps right, 2 steps down, and 5 steps forward."

Key Takeaway:

A vector is a journey from one place to another. It doesn't matter where you start; it only matters how far and in what direction you travel.

2. Magnitude and Direction

Sometimes we need to know exactly how long a vector is. This is called the magnitude.

Calculating Magnitude

Think of this as an extension of Pythagoras’ Theorem. To find the magnitude of vector \(\mathbf{a} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k}\), we use the notation \(|\mathbf{a}|\):

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

Example: Find the magnitude of \(\mathbf{v} = \begin{pmatrix} 3 \\ -4 \\ 0 \end{pmatrix}\).
\(|\mathbf{v}| = \sqrt{3^2 + (-4)^2 + 0^2} = \sqrt{9 + 16} = \sqrt{25} = 5\).

Finding Direction (2D)

In 2D, the direction is usually given as an angle with the positive \(x\)-axis (the \(\mathbf{i}\) direction). Use trigonometry:
\(\tan(\theta) = \frac{y}{x}\)

Common Mistake: When calculating the angle, always sketch the vector first! If your vector is in the "bottom-left" quadrant, your calculator might give you a misleading answer. A quick sketch saves marks!

Key Takeaway:

Magnitude is the "length" of the arrow. Use Pythagoras. Direction is the "angle." Use \(\tan^{-1}\).

3. Adding, Subtracting, and Scaling

Working with vectors is very similar to basic algebra, but with a few geometric rules.

Vector Addition

Algebraically: Just add the components together.
\(\begin{pmatrix} 2 \\ 3 \end{pmatrix} + \begin{pmatrix} 4 \\ -1 \end{pmatrix} = \begin{pmatrix} 2+4 \\ 3+(-1) \end{pmatrix} = \begin{pmatrix} 6 \\ 2 \end{pmatrix}\)

Diagrammatically: Use the "Tip-to-Tail" method. Place the start of the second vector at the end of the first one. The "Resultant" vector is the shortcut from the very start to the very end.

Scalar Multiplication

If you multiply a vector by a regular number (a scalar), you just make it longer or shorter.
\(2 \times \begin{pmatrix} 3 \\ -1 \end{pmatrix} = \begin{pmatrix} 6 \\ -2 \end{pmatrix}\)

Did you know? If two vectors are parallel, one will always be a scalar multiple of the other. For example, \(\begin{pmatrix} 1 \\ 2 \end{pmatrix}\) and \(\begin{pmatrix} 5 \\ 10 \end{pmatrix}\) are parallel because the second one is just \(5 \times\) the first one.

Key Takeaway:

Add components to add vectors. If one vector is a multiple of another, they are pointing in the same (or exactly opposite) direction.

4. Position Vectors and Distance

A Position Vector is a vector that starts at the origin \(O (0,0,0)\). We usually write the position of point \(A\) as \(\vec{OA}\) or \(\mathbf{a}\).

The Journey Between Two Points

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

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

Memory Aid: It's always "Finish minus Start." To get from \(A\) to \(B\), you subtract the position of where you started from where you ended up.

Distance Between Two Points

To find the distance between point \(A\) and point \(B\), simply find the vector \(\vec{AB}\) and then calculate its magnitude.

Key Takeaway:

\(\vec{AB} = \text{Position } B - \text{Position } A\). The distance is the magnitude of this result.

5. Vectors in the Real World (Mechanics)

Since this is Paper 2, you will likely see vectors used in Mechanics (Forces and Kinematics).

Velocity and Displacement

  • Displacement is a vector (where you are relative to the start).
  • Velocity is a vector (how fast you are moving and in what direction).
  • Speed is a scalar (it is just the magnitude of the velocity vector).

Resultant Forces

When multiple forces act on an object, the "total" force is called the Resultant Force. You find it by adding all the individual force vectors together.
If the forces add up to \(\begin{pmatrix} 0 \\ 0 \end{pmatrix}\), the object is in equilibrium (it's not accelerating!).

Analogy: Imagine two people pulling a rope in different directions. The direction the rope actually moves depends on the combined "vector sum" of their pulls.

Key Takeaway:

In mechanics, "Magnitude" usually means "Speed" (for velocity) or "Total Force" (for force vectors).

Quick Summary Checklist

  • Can you convert between \(\mathbf{i}, \mathbf{j}, \mathbf{k}\) and column vectors?
  • Do you remember to use \(\sqrt{x^2+y^2+z^2}\) for magnitude?
  • Are you using "Finish minus Start" for \(\vec{AB}\)?
  • Can you show two vectors are parallel by finding a common factor?

Final Tip: Don't let the 3D aspect scare you! The math for 3D vectors is exactly the same as 2D; you just have one extra number (\(z\)) to include in your sums. You've got this!