M2 Algebra and Calculus: Introduction to Vectors
Hey everyone! Welcome to your study notes for one of the most visual and practical topics in M2: Vectors. Don't worry if the name sounds a bit intimidating. Vectors are just a cool way to talk about quantities that have both a size and a direction. Think about giving directions: "walk 500 metres north". The '500 metres' is the size, and 'north' is the direction. That's a vector!
In this chapter, we'll learn the language of vectors, how to add and subtract them, and how to work with them in a coordinate system. This is super useful in physics, computer graphics, engineering, and even video game design!
1. What are Vectors and Scalars?
In math and science, we use two types of quantities to describe the world:
Scalars: Just a Number
A scalar is a quantity that has only magnitude (which is just a fancy word for 'size' or 'amount').
Examples:
Your height (e.g., 1.7 metres)
The temperature (e.g., 25°C)
The speed of a car (e.g., 50 km/h)
How much money you have (e.g., $100)
These are all just numbers. They don't have a direction.
Vectors: A Number AND a Direction
A vector is a quantity that has both magnitude AND direction. This is the key difference!
Examples:
Velocity: A car travelling at 50 km/h East.
Force: Pushing a box with a force of 20 Newtons downwards.
Displacement: Walking 1 km North from your home.
How We Write and Draw Vectors
Drawing vectors is easy! We just use an arrow. The length of the arrow shows its magnitude, and where it points shows its direction.
Vector Notation:
In textbooks, you'll see vectors as bold letters, like a or v.
When you write them by hand, you should put an arrow on top, like $$ \vec{a} $$ or $$ \vec{v} $$.
A vector from a point A to a point B is written as $$ \vec{AB} $$.
Magnitude Notation:
The magnitude (or length) of a vector a is written as $$ |\textbf{a}| $$ or $$ |\vec{a}| $$. The magnitude is always a positive number or zero (it's a scalar!).
Special Types of Vectors
Zero Vector: This is a vector with zero magnitude and no specific direction. It's written as 0 or $$ \vec{0} $$. Think of it as the instruction "don't move".
Unit Vector: This is a very important concept! A unit vector is any vector that has a magnitude of exactly 1. Unit vectors are used to describe a direction. A unit vector in the direction of vector a is often written as $$ \hat{\textbf{a}} $$.
Key Takeaway
Scalars are just numbers (magnitude). Vectors have both magnitude and direction. Remember: Velocity is a vector, but speed is a scalar!
2. Vector Operations: The Basic Rules
Just like with numbers, we can perform operations like addition, subtraction, and multiplication with vectors. It's a bit different, but the logic is simple!
Vector Addition
How do we add two vectors, say $$ \vec{a} $$ and $$ \vec{b} $$? We can think of it as following two sets of directions one after the other.
The Triangle Law (or "Tip-to-Tail" Method):
Draw the first vector, $$ \vec{a} $$.
Draw the second vector, $$ \vec{b} $$, starting from the tip (the arrow end) of $$ \vec{a} $$.
The resulting vector, $$ \vec{a} + \vec{b} $$, is the arrow drawn from the tail (start) of $$ \vec{a} $$ to the tip (end) of $$ \vec{b} $$.
Analogy: Imagine you walk East for 3 km ($$ \vec{a} $$), and then you walk North for 4 km ($$ \vec{b} $$). The vector $$ \vec{a} + \vec{b} $$ represents the direct path from your starting point to your final destination.
Vector Subtraction
To subtract a vector, we just add its negative. But what's a negative vector?
The vector $$ -\vec{b} $$ is a vector with the same magnitude as $$ \vec{b} $$ but pointing in the exact opposite direction.
So, to calculate $$ \vec{a} - \vec{b} $$, we simply do $$ \vec{a} + (-\vec{b}) $$. We just flip the direction of $$ \vec{b} $$ and then use the tip-to-tail method!
Scalar Multiplication
This means multiplying a vector by a scalar (a regular number). Let's say we have a vector a and a scalar k.
The vector ka has a direction that is the same as a if k is positive.
The vector ka has a direction that is opposite to a if k is negative.
The magnitude of ka is $$|k| |\textbf{a}| $$. We just multiply the original magnitude by the absolute value of the scalar.
Example: The vector 2a points in the same direction as a but is twice as long. The vector -0.5a points in the opposite direction and is half as long.
Properties of Vector Operations
Good news! Vector operations follow some very familiar rules. For any vectors a, b, c and scalars λ, μ:
$$ \textbf{a} + \textbf{b} = \textbf{b} + \textbf{a} $$ (Order of addition doesn't matter)
$$ (\textbf{a} + \textbf{b}) + \textbf{c} = \textbf{a} + (\textbf{b} + \textbf{c}) $$ (Grouping of addition doesn't matter)
$$ \textbf{a} + \textbf{0} = \textbf{a} $$ (Adding the zero vector changes nothing)
$$ (\lambda + \mu)\textbf{a} = \lambda\textbf{a} + \mu\textbf{a} $$ (Distributive law for scalars)
$$ \lambda(\textbf{a} + \textbf{b}) = \lambda\textbf{a} + \lambda\textbf{b} $$ (Distributive law for vectors)
$$ \lambda(\mu\textbf{a}) = (\lambda\mu)\textbf{a} $$ (Grouping of scalar multiplication doesn't matter)
Key Takeaway
Adding vectors is like following paths one after another (tip-to-tail). Subtracting is just adding the opposite vector. Multiplying by a scalar stretches, shrinks, and/or flips the vector's direction.
3. Vectors in the Coordinate System
Drawing vectors is great for understanding concepts, but it's not very precise for calculations. That's where the coordinate system (x-y plane or x-y-z space) comes in. This makes everything much easier!
Representing Vectors with Coordinates
We can represent any vector using its components along the x, y, and z axes. We usually write this as a column vector.
A vector a in 2D is: $$ \textbf{a} = \begin{pmatrix} a_x \\ a_y \end{pmatrix} $$
A vector b in 3D is: $$ \textbf{b} = \begin{pmatrix} b_x \\ b_y \\ b_z \end{pmatrix} $$
Position Vectors
A position vector is a vector that starts at the origin (0, 0) or (0, 0, 0) and ends at a specific point P. The position vector of point P(x, y, z) is simply $$ \vec{OP} = \begin{pmatrix} x \\ y \\ z \end{pmatrix} $$.
Vector Between Two Points
This is a very common task! To find the vector $$ \vec{AB} $$ that starts at point A and ends at point B, we use their position vectors:
Formula: $$ \vec{AB} = \vec{OB} - \vec{OA} $$
In simple terms: "End point minus Start point"
Example: Find the vector from A(1, 2) to B(5, 8).
$$ \vec{OA} = \begin{pmatrix} 1 \\ 2 \end{pmatrix}, \vec{OB} = \begin{pmatrix} 5 \\ 8 \end{pmatrix} $$
$$ \vec{AB} = \vec{OB} - \vec{OA} = \begin{pmatrix} 5 \\ 8 \end{pmatrix} - \begin{pmatrix} 1 \\ 2 \end{pmatrix} = \begin{pmatrix} 5-1 \\ 8-2 \end{pmatrix} = \begin{pmatrix} 4 \\ 6 \end{pmatrix} $$
Common Mistake Alert!
Be careful! $$ \vec{AB} $$ is NOT the same as $$ \vec{BA} $$. $$ \vec{BA} = \vec{OA} - \vec{OB} = - \vec{AB} $$. They have the same length but point in opposite directions.
Operations in Coordinate Form
This is the easy part. You just apply the operations to each component separately.
Let $$ \textbf{a} = \begin{pmatrix} a_x \\ a_y \end{pmatrix} $$, $$ \textbf{b} = \begin{pmatrix} b_x \\ b_y \end{pmatrix} $$, and k be a scalar.
Addition: $$ \textbf{a} + \textbf{b} = \begin{pmatrix} a_x + b_x \\ a_y + b_y \end{pmatrix} $$
Subtraction: $$ \textbf{a} - \textbf{b} = \begin{pmatrix} a_x - b_x \\ a_y - b_y \end{pmatrix} $$
Scalar Multiplication: $$ k\textbf{a} = \begin{pmatrix} ka_x \\ ka_y \end{pmatrix} $$
(The same rules apply for 3D vectors, just with an extra z-component!)
Finding the Magnitude in Coordinates
This is just Pythagoras' Theorem!
For a 2D vector $$ \textbf{a} = \begin{pmatrix} x \\ y \end{pmatrix} $$, the magnitude is:
$$ |\textbf{a}| = \sqrt{x^2 + y^2} $$For a 3D vector $$ \textbf{b} = \begin{pmatrix} x \\ y \\ z \end{pmatrix} $$, the magnitude is:
$$ |\textbf{b}| = \sqrt{x^2 + y^2 + z^2} $$Example: Find the magnitude of $$ \vec{v} = \begin{pmatrix} 3 \\ 4 \end{pmatrix} $$
$$ |\vec{v}| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 $$
Did you know?
In 2D, the components of a vector can be related to its magnitude and angle. If a vector $$ \vec{v} $$ has magnitude $$ |\vec{v}| $$ and makes an angle θ with the positive x-axis, then its components are $$ x = |\vec{v}|\cos\theta $$ and $$ y = |\vec{v}|\sin\theta $$. This is the same as the trigonometry you learned for circles!
Key Takeaway
Using coordinates turns vector problems into simple arithmetic. Remember the key formulas: $$ \vec{AB} = \vec{OB} - \vec{OA} $$ and $$ |\vec{v}| = \sqrt{x^2 + y^2 + z^2} $$.
4. A Key Condition for Comparing Vectors
There's one more important property from the syllabus you need to know. It's about when two combinations of vectors are equal.
The rule is: If you have two vectors, a and b, which are non-zero and not parallel to each other...
...and you are told that:
$$ \alpha\textbf{a} + \beta\textbf{b} = \alpha'\textbf{a} + \beta'\textbf{b} $$...then it MUST be true that the coefficients are equal:
$$ \alpha = \alpha' \quad \text{and} \quad \beta = \beta' $$Why is this important? Think of a and b as being like independent directions, for example 'steps East' and 'steps North'. If two people start at the same point and end up at the same final location by only taking East and North steps, they must have taken the same number of East steps and the same number of North steps. You can't substitute East steps for North steps and end up in the same place!
Example: If $$ (k+1)\textbf{a} + 5\textbf{b} = 4\textbf{a} + (m-2)\textbf{b} $$, where a and b are not parallel, find k and m.
By comparing the coefficients:
For vector a: $$ k+1 = 4 \implies k = 3 $$
For vector b: $$ 5 = m-2 \implies m = 7 $$
And that's it! You've got the foundational concepts of vectors. Practice using them in coordinate form, and you'll find they are a powerful and simple tool to solve many problems.