Welcome to the World of Vectors!
Hey there! Today, we are going to dive into the world of Vectors. If you’ve ever followed directions like "walk three blocks east and two blocks north," you’ve already used vectors without even knowing it!
In your Oxford AQA International AS Level Mathematics course, vectors appear in both Pure Math (for shifting graphs) and Mechanics (for forces and movement). Don't worry if it sounds a bit "physics-heavy" at first—we’re going to break it down into simple, bite-sized pieces that make sense.
1. What exactly is a Vector?
In math, we usually deal with Scalars. A scalar is just a number, like "5 kg" or "10 degrees Celsius." It tells us how much, but not which way.
A Vector, however, has two parts: 1. Magnitude (how big it is or how far it goes). 2. Direction (which way it is pointing).
Analogy: Imagine you are a captain of a ship. If you tell your crew to "sail at 20 knots," they won't know where to go (Scalar). If you say "sail at 20 knots toward the North Star," now they have a Vector!
Quick Review: - Scalar: Size only (e.g., Distance, Speed). - Vector: Size AND Direction (e.g., Displacement, Velocity, Force).
2. How we write Vectors (Notation)
In your exams, you will mostly see vectors written in two ways:
A. Column Vectors
This is the most common format you'll see in the P1 syllabus. It looks like a tall bracket with two numbers:
\( \mathbf{v} = \begin{bmatrix} x \\ y \end{bmatrix} \)
- 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).
B. Letter Notation
In textbooks, vectors are often printed in bold (like \(\mathbf{a}\)). Since you can't write in bold with a pen, you should underline your vectors (like \( \underline{a} \)). You might also see them written as the path between two points, like \( \vec{AB} \).
Key Takeaway: Always think of a column vector as a set of instructions: "Go across, then go up/down."
3. Adding and Subtracting Vectors
The great news is that adding and subtracting vectors is as simple as basic addition! You just handle the top numbers and bottom numbers separately.
Addition: If you move by vector \(\mathbf{a}\) and then move by vector \(\mathbf{b}\), your total movement is \( \mathbf{a} + \mathbf{b} \).
Example: \( \begin{bmatrix} 2 \\ 3 \end{bmatrix} + \begin{bmatrix} 4 \\ -1 \end{bmatrix} = \begin{bmatrix} 2+4 \\ 3+(-1) \end{bmatrix} = \begin{bmatrix} 6 \\ 2 \end{bmatrix} \)
Subtraction: To subtract, just subtract the corresponding numbers.
Example: \( \begin{bmatrix} 5 \\ 10 \end{bmatrix} - \begin{bmatrix} 2 \\ 3 \end{bmatrix} = \begin{bmatrix} 3 \\ 7 \end{bmatrix} \)
Multiplying by a Scalar: If you multiply a vector by a normal number, it just makes the vector longer or shorter (or flips it if the number is negative).
Example: \( 3 \times \begin{bmatrix} 2 \\ -1 \end{bmatrix} = \begin{bmatrix} 6 \\ -3 \end{bmatrix} \)
Common Mistake: Don't accidentally add the top number to the bottom number! Keep the \(x\) and \(y\) worlds completely separate.
4. Finding the "Size" (Magnitude)
Sometimes the question will ask for the magnitude of a vector. This is just a fancy way of asking "How long is the arrow?"
Because a vector forms a right-angled triangle (across and up), we use our old friend Pythagoras’ Theorem!
The magnitude of vector \( \begin{bmatrix} x \\ y \end{bmatrix} \) is written as \( |\mathbf{v}| \).
Formula: \( |\mathbf{v}| = \sqrt{x^2 + y^2} \)
Example: Find the magnitude of \( \begin{bmatrix} 3 \\ 4 \end{bmatrix} \).
\( \text{Magnitude} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \).
5. Vectors in Graph Transformations
In the P1 Algebra section of your syllabus, you are required to use vectors to describe Translations of graphs.
If you have a graph \( y = f(x) \) and you move it using the vector \( \begin{bmatrix} a \\ b \end{bmatrix} \): - The graph shifts \(a\) units to the right (if \(a\) is positive). - The graph shifts \(b\) units up (if \(b\) is positive).
Did you know? In the equation of the new graph, the \(x\) shift feels "backwards." A translation of \( \begin{bmatrix} 3 \\ -2 \end{bmatrix} \) changes \( f(x) \) into \( f(x-3) - 2 \). Notice how the \(+3\) in the vector becomes \(-3\) inside the bracket!
Key Takeaway: A vector \( \begin{bmatrix} 3 \\ -2 \end{bmatrix} \) means "Shift the whole graph 3 units right and 2 units down."
6. Vectors in Mechanics (Unit PSM1)
In the Mechanics part of your course, vectors represent things like Force and Velocity.
Resultant Forces: If two people are pulling an object with different forces (vectors), the "Resultant Force" is simply the sum of those two vectors.
Step-by-Step for Resultant Forces:
1. Write each force as a column vector \( \begin{bmatrix} x \\ y \end{bmatrix} \).
2. Add the vectors together to get the resultant vector.
3. If the question asks for the "magnitude of the force," use \( \sqrt{x^2 + y^2} \).
4. If the question asks for the "direction," use trigonometry (\( \tan \theta = \frac{y}{x} \)).
Summary Checklist
Before you finish this chapter, make sure you can: - [ ] Write a vector as a column vector \( \begin{bmatrix} x \\ y \end{bmatrix} \). - [ ] Add and subtract vectors by combining the top and bottom numbers. - [ ] Calculate the magnitude using \( \sqrt{x^2 + y^2} \). - [ ] Use a vector to translate a graph (e.g., shifting \( y = x^2 \) to the right and down). - [ ] Understand that a negative vector (like \( -\mathbf{a} \)) just points in the opposite direction.
Encouraging Note: Vectors can feel strange because they are "two numbers in one," but once you realize you just treat the top and bottom separately, they become one of the most straightforward parts of the syllabus. Keep practicing!