Welcome to the World of Vectors!
In this chapter, we are going to explore Vectors. If you’ve ever followed a map or played a video game where a character moves across the screen, you’ve already encountered vectors! Unlike regular numbers (which we call scalars), vectors don't just tell us "how much"—they also tell us "which way."
By the end of these notes, you’ll be able to navigate 2D and 3D space, calculate distances, and solve geometric puzzles using vector algebra. 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 5 kg or 10 minutes.
A vector has both magnitude (size) and direction.
Analogy: If I tell you "Walk 5 miles," you might end up anywhere (that's a scalar). If I tell you "Walk 5 miles North," you know exactly where to go (that's a vector)!
Representing Vectors
We usually write vectors in two ways:
1. Column Vectors: Written vertically. In 2D: \( \begin{pmatrix} x \\ y \end{pmatrix} \). In 3D: \( \begin{pmatrix} x \\ y \\ z \end{pmatrix} \).
2. Unit Vector Notation: Using i, j, and k. These are "basis vectors" that represent 1 unit in the x, y, and z directions respectively.
Example: A vector moving 3 units right, 4 units up, and 2 units "out" (in 3D) is written as:
\( \mathbf{a} = 3\mathbf{i} + 4\mathbf{j} + 2\mathbf{k} \) or \( \begin{pmatrix} 3 \\ 4 \\ 2 \end{pmatrix} \).
Did you know? In textbooks, vectors are usually bold (like a). When you write them by hand, you should underline them (like a) so your examiner knows they aren't just regular numbers!
Key Takeaway: Vectors represent a movement or a position in space using coordinates (x, y, z) or unit vectors (\(\mathbf{i}, \mathbf{j}, \mathbf{k}\)).
2. Magnitude and Direction
The magnitude of a vector is simply its length. We use the notation \( |\mathbf{a}| \) to represent this.
Calculating Magnitude
To find the length, we use a 3D version of Pythagoras’ Theorem.
For a vector \( \mathbf{a} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k} \):
\( |\mathbf{a}| = \sqrt{x^2 + y^2 + z^2} \)
Example: Find the magnitude of \( \mathbf{v} = 3\mathbf{i} - 2\mathbf{j} + 6\mathbf{k} \).
\( |\mathbf{v}| = \sqrt{3^2 + (-2)^2 + 6^2} = \sqrt{9 + 4 + 36} = \sqrt{49} = 7 \).
Unit Vectors
A unit vector is a vector with a magnitude of 1. If you want to find a unit vector in the same direction as a, you just divide a by its own magnitude. We call this "normalising" the vector, written as \( \hat{\mathbf{a}} \).
\( \hat{\mathbf{a}} = \frac{\mathbf{a}}{|\mathbf{a}|} \)
Direction in 2D
In 2D, we often describe direction as an angle \( \theta \) from the positive x-axis.
Using trigonometry: \( \tan \theta = \frac{y}{x} \). Always draw a quick sketch to check which quadrant your angle is in!
Quick Review:
- Magnitude = Length of the arrow.
- Unit Vector = Length of exactly 1.
- Use Pythagoras for length and Trig for direction.
3. Vector Addition and Scalar Multiplication
Working with vectors is very similar to basic algebra, but with a few geometric rules.
Addition and Subtraction
To add or subtract vectors algebraically, just add/subtract their components.
If \( \mathbf{a} = \begin{pmatrix} 1 \\ 2 \end{pmatrix} \) and \( \mathbf{b} = \begin{pmatrix} 3 \\ -1 \end{pmatrix} \), then \( \mathbf{a} + \mathbf{b} = \begin{pmatrix} 1+3 \\ 2+(-1) \end{pmatrix} = \begin{pmatrix} 4 \\ 1 \end{pmatrix} \).
Geometrical Interpretation:
1. Triangle Law: Place the start of the second vector at the end of the first. The "shortcut" from start to finish is the resultant vector.
2. Parallelogram Law: Place both vectors starting from the same point. The diagonal of the parallelogram they form is the sum.
Scalar Multiplication
When you multiply a vector by a number (a scalar), you change its length but not its direction (unless the number is negative, which flips the direction).
Example: \( 2 \times \begin{pmatrix} 3 \\ 4 \end{pmatrix} = \begin{pmatrix} 6 \\ 8 \end{pmatrix} \).
Parallel Vectors
Two vectors are parallel if one is a scalar multiple of the other.
If \( \mathbf{a} = k\mathbf{b} \), then a and b are parallel.
Common Mistake to Avoid: When subtracting vectors, like \( \mathbf{a} - \mathbf{b} \), remember it's the same as \( \mathbf{a} + (-\mathbf{b}) \). You are adding the "reverse" of vector b.
Key Takeaway: Adding vectors is like following a path. Multiplying by a scalar "stretches" or "shrinks" that path.
4. Position Vectors and Distance
A Position Vector tells us the location of a point relative to a fixed origin, O (usually \( (0,0,0) \)). We write the position of point A as \( \vec{OA} \) or simply \( \mathbf{a} \).
The "B minus A" Rule
This is one of the most important tools in your vector toolkit! If you want to find the vector that takes you from point A to point B (\( \vec{AB} \)), you use:
\( \vec{AB} = \vec{OB} - \vec{OA} \) (or simply \( \mathbf{b} - \mathbf{a} \)).
Memory Aid: To get from A to B, it’s "Finish minus Start."
Distance Between Two Points
The distance between point \( (x_1, y_1, z_1) \) and point \( (x_2, y_2, z_2) \) is just the magnitude of the vector \( \vec{AB} \).
\( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \)
Quick Review Box:
- \( \vec{OA} \): Where A is relative to the origin.
- \( \vec{AB} = \mathbf{b} - \mathbf{a} \): How to get from A to B.
- Distance = Magnitude of \( \vec{AB} \).
5. Solving Geometric Problems
You can use vectors to prove things about shapes, like whether a quadrilateral is a parallelogram.
Example: Finding the 4th corner of a parallelogram
If you have a parallelogram ABCD, then the vector \( \vec{AB} \) must be exactly the same as the vector \( \vec{DC} \) because they are parallel and the same length.
Step-by-Step:
1. Find the vector \( \vec{AB} \) using \( \mathbf{b} - \mathbf{a} \).
2. Set this equal to \( \vec{DC} \), which is \( \mathbf{c} - \mathbf{d} \).
3. Solve for the missing coordinates of d.
Vectors in Context
In Pure Mathematics papers, you might see vectors used to describe forces or velocities.
- The Resultant Force is just the sum of all individual force vectors.
- If an object is in equilibrium, the sum of the vectors is zero \( \begin{pmatrix} 0 \\ 0 \end{pmatrix} \).
Don't worry if this seems tricky at first! Drawing a diagram is almost always the best way to start. Once you see the "arrows," the math usually follows!
Final Takeaway: Vectors are just "directions with distances." Whether you are in 2D or 3D, the rules of adding, subtracting, and finding lengths stay exactly the same!