Vectors

Introduction to Vectors

In your GCSE journey, you have already used coordinates to describe a fixed point on a grid. Vectors are slightly different; instead of telling us where something is, they tell us how to move from one place to another. Vectors are used everywhere in the real world, from describing the path of an airplane to the forces acting on a bridge.

Think of a vector as a set of instructions: "Go 3 steps East and 2 steps North." It doesn't matter where you start; if you follow those instructions, you have followed that vector!

1. Understanding Vector Notation

There are three main ways you will see vectors written in your exam:

1. Bold or Underlined Letters: We often name a vector using a single lowercase letter. In printed textbooks, it will be bold, like \( \mathbf{a} \). Since you cannot write in bold during an exam, you should underline it, like \( \underline{a} \).

2. Directed Line Segments: If a vector goes from point \( A \) to point \( B \), we write it as \( \vec{AB} \). The arrow shows the direction of travel.

3. Column Vectors: This is the most common way to do calculations. A column vector looks like this: \( \begin{pmatrix} x \\ y \end{pmatrix} \).
- The top number (\( x \)) tells you the horizontal movement (right is positive, left is negative).
- The bottom number (\( y \)) tells you the vertical movement (up is positive, down is negative).

Quick Review:

\( \begin{pmatrix} 3 \\ -2 \end{pmatrix} \) means: Move 3 units to the right and 2 units down.

2. Vector Arithmetic

Don't worry if this seems like a lot of new symbols! Calculating with vectors is actually very straightforward. It is just like basic addition and multiplication, but you do it in two layers (top and bottom).

Addition and Subtraction

To add or subtract vectors, simply add or subtract the corresponding numbers in the columns.

Example: If \( \mathbf{a} = \begin{pmatrix} 4 \\ 1 \end{pmatrix} \) and \( \mathbf{b} = \begin{pmatrix} 2 \\ 5 \end{pmatrix} \), then:
\( \mathbf{a} + \mathbf{b} = \begin{pmatrix} 4 + 2 \\ 1 + 5 \end{pmatrix} = \begin{pmatrix} 6 \\ 6 \end{pmatrix} \)

Visualizing it: Adding vectors is like following two sets of instructions one after the other. If you go along vector \( \mathbf{a} \) and then along vector \( \mathbf{b} \), the "shortcut" from your start to your finish is the resultant vector \( \mathbf{a} + \mathbf{b} \).

Multiplication by a Scalar

A "scalar" is just a normal number. When you multiply a vector by a scalar, you multiply both the top and bottom numbers by that value.

Example: If \( \mathbf{a} = \begin{pmatrix} 3 \\ -1 \end{pmatrix} \), then \( 2\mathbf{a} = \begin{pmatrix} 2 \times 3 \\ 2 \times -1 \end{pmatrix} = \begin{pmatrix} 6 \\ -2 \end{pmatrix} \).

Key Concept: The vector \( 2\mathbf{a} \) is in the same direction as \( \mathbf{a} \), but it is twice as long. If you multiply by a negative number, like \( -\mathbf{a} \), the vector stays the same length but points in the exact opposite direction.

3. Diagrammatic Representations

In exam questions, you will often be given a grid or a shape with arrows.
- Parallel Vectors: If one vector is a multiple of another (like \( \mathbf{a} \) and \( 3\mathbf{a} \)), they are parallel.
- Negative Vectors: The vector \( \vec{BA} \) is the negative of \( \vec{AB} \). If \( \vec{AB} = \mathbf{a} \), then \( \vec{BA} = -\mathbf{a} \).

Common Mistake to Avoid:

When drawing a resultant vector like \( \mathbf{a} - \mathbf{b} \), remember that this is the same as \( \mathbf{a} + (-\mathbf{b}) \). Go along \( \mathbf{a} \), then go in the opposite direction of \( \mathbf{b} \).

4. Geometric Arguments and Proofs (Higher Tier Only)

Higher tier students need to use vectors to prove geometric facts. This usually involves finding a "path" from one point to another using known vectors.

Proving Lines are Parallel

To prove two lines are parallel, you must show that their vectors are multiples of each other.

Example: If \( \vec{PQ} = 2\mathbf{a} + \mathbf{b} \) and \( \vec{RS} = 6\mathbf{a} + 3\mathbf{b} \), you can factorise the second one:
\( \vec{RS} = 3(2\mathbf{a} + \mathbf{b}) \).
Since \( \vec{RS} = 3 \times \vec{PQ} \), the lines are parallel.

Proving Points Lie on a Straight Line (Collinear)

To prove that three points \( A \), \( B \), and \( C \) lie on a straight line:
1. Find the vector \( \vec{AB} \).
2. Find the vector \( \vec{BC} \) (or \( \vec{AC} \)).
3. Show that one is a multiple of the other (proving they are parallel).
4. State that because they both pass through a common point (like \( B \)), they must be on the same straight line.

Key Takeaway for Proofs:

Always look for midpoints or ratios in the question. If \( M \) is the midpoint of \( \vec{AB} \), then \( \vec{AM} = \frac{1}{2}\vec{AB} \).

Summary: Vectors Quick Check

• Column Vectors: Top is Right/Left, Bottom is Up/Down.
• Addition: Add the components. Diagram-wise, it’s "tip-to-tail".
• Scalar Multiplication: Multiplies the length; negatives reverse the direction.
• (Higher) Parallel: One vector is a scalar multiple of the other.
• (Higher) Collinear: Parallel vectors that share a common point.

Remember: Vectors are just a way of describing a journey. If you can find a path from the start point to the end point by following the arrows, you can solve any vector problem!