Welcome to the World of Vectors!
Welcome to one of the most exciting and visual chapters in H2 Mathematics! If you’ve ever used a GPS to find your way or played a 3D video game, you’ve already encountered vectors. In this chapter, we will learn how to describe movement and position in space using arrows.
Think of a vector as a set of instructions: "Go 3 km North and 4 km East." Unlike a regular number (like 5 kg), which only tells us "how much," a vector tells us "how much" (magnitude) and "which way" (direction).
1. Understanding the Basics: What is a Vector?
A vector is a quantity that has both magnitude (size) and direction. In your exams, you will see them written in a few ways:
- Bold letters: a
- Underlined letters: a (this is how you should write them in your paper!)
- As a column: \( \begin{pmatrix} x \\ y \\ z \end{pmatrix} \)
Addition and Subtraction
Imagine walking from point A to point B (vector a), and then from B to C (vector b). The total journey from A to C is the vector sum: \( \mathbf{a} + \mathbf{b} \).
The Triangle Law: To add vectors, place the "tail" of the second vector at the "head" of the first. The result is the arrow that closes the triangle.
Vector Subtraction: To find \( \mathbf{a} - \mathbf{b} \), you can think of it as \( \mathbf{a} + (-\mathbf{b}) \). Simply flip the direction of b and add it to a.
Multiplication by a Scalar
A "scalar" is just a fancy word for a normal number (like 2, 5, or -1). When you multiply a vector by a scalar \( k \):
- If \( k > 1 \), the vector gets longer (stretched).
- If \( 0 < k < 1 \), the vector gets shorter (compressed).
- If \( k \) is negative, the vector flips direction.
Quick Review: Adding vectors is like combining two movements. Multiplying by a scalar changes the length but keeps the direction (or reverses it).
2. Position, Displacement, and Direction Vectors
This is where many students get confused, but the difference is actually quite simple!
Position Vectors
A position vector always starts from the Origin (O), which is the point \( (0,0,0) \). It tells you exactly where a point is located in space.
Example: The position vector of point \( A \) is \( \vec{OA} = \mathbf{a} \).
Displacement Vectors
A displacement vector tells you how to get from one point to another point (neither of which have to be the origin).
The Golden Rule: To find the vector from A to B (\( \vec{AB} \)), use the formula:
\( \vec{AB} = \vec{OB} - \vec{OA} \)
Memory Aid: Think "Back minus Front." To find \( \vec{AB} \), take the position of the second letter (B) and subtract the position of the first letter (A).
Direction Vectors
A direction vector just shows the "slope" or "aim" of a line. It doesn't care where you start; it only cares which way you are pointing.
3. Magnitude and Unit Vectors
Sometimes we only care about how long a vector is. This is called the magnitude.
Calculating Magnitude
In 3D, if \( \mathbf{v} = \begin{pmatrix} x \\ y \\ z \end{pmatrix} \), the magnitude is found using a 3D version of Pythagoras' Theorem:
\( |\mathbf{v}| = \sqrt{x^2 + y^2 + z^2} \)
Unit Vectors
A unit vector is a vector that has a magnitude of exactly 1 unit. We often use it to indicate direction without changing the size of other quantities.
To turn any vector \( \mathbf{v} \) into a unit vector (written as \( \hat{\mathbf{v}} \)), simply divide the vector by its own magnitude:
\( \hat{\mathbf{v}} = \frac{\mathbf{v}}{|\mathbf{v}|} \)
Did you know? The standard unit vectors are \( \mathbf{i}, \mathbf{j}, \) and \( \mathbf{k} \). They represent 1 unit of movement in the x, y, and z directions respectively!
4. Distance Between Two Points
If you have two points A and B, the distance between them is simply the magnitude of the displacement vector \( \vec{AB} \).
Step-by-Step:
1. Find \( \vec{AB} \) using \( \vec{OB} - \vec{OA} \).
2. Calculate the magnitude \( |\vec{AB}| \) using the square root formula.
5. Collinearity: Points on a Line
Three points A, B, and C are collinear if they all lie on the same straight line.
To prove this, you need to show two things:
1. The vectors \( \vec{AB} \) and \( \vec{BC} \) are parallel. (This means \( \vec{AB} = k\vec{BC} \) for some number \( k \)).
2. They share a common point (like point B).
6. The Ratio Theorem
Don't worry if this seems tricky at first! The Ratio Theorem is just a formula to find a point that divides a line segment into a specific ratio.
If point \( P \) divides the line \( AB \) in the ratio \( m : n \), the position vector \( \mathbf{p} \) is:
\( \mathbf{p} = \frac{n\mathbf{a} + m\mathbf{b}}{m + n} \)
The "Criss-Cross" Trick: Notice that the \( n \) (which is next to \( \mathbf{a} \) in the ratio) gets multiplied by \( \mathbf{a} \), and the \( m \) (which is next to \( \mathbf{b} \)) gets multiplied by \( \mathbf{b} \). It’s an "opposite" multiplication!
Special Case: The Midpoint
For a midpoint, the ratio is \( 1 : 1 \). The formula becomes much simpler:
\( \mathbf{p} = \frac{\mathbf{a} + \mathbf{b}}{2} \)
Summary: Key Takeaways
- Vector \( \vec{AB} \): Always \( \text{Position B} - \text{Position A} \).
- Magnitude: Use \( \sqrt{x^2 + y^2 + z^2} \).
- Unit Vector: The original vector divided by its length.
- Parallel Vectors: One is a multiple of the other (\( \mathbf{a} = k\mathbf{b} \)).
- Ratio Theorem: Use the "criss-cross" method to find internal points.
Common Mistake to Avoid: Forgetting to underline your vectors in the exam! In print they are bold, but on paper you must write a or use the arrow \( \vec{OA} \) to show it's a vector and not a scalar.