🧭 Study Notes: Vectors in Two Dimensions (International Mathematics 0607)

Hey there, future mathematicians! Welcome to the world of vectors. This chapter is super important because it connects algebra with geometry, giving us a powerful tool to describe movement and forces accurately. Don't worry if this seems tricky at first; we’ll break it down into simple, manageable steps!

What is a Vector? A Simple Explanation

In maths, quantities can be split into two types:

  • Scalar Quantities: These only have magnitude (size).
    Examples: Speed (40 km/h), Distance (10 meters), Time (5 seconds).
  • Vector Quantities: These have both magnitude (size) AND direction.
    Examples: Velocity (40 km/h North), Displacement (10 meters East), Force (5 Newtons upwards).

Analogy: Imagine you are searching for buried treasure.

  • If someone tells you, "Walk 10 metres," that's a scalar (distance). You don't know which way to go!
  • If someone tells you, "Walk 10 metres due North," that's a vector (displacement). You know exactly where to end up!
Vector Notation

In exams, vectors are often represented in two ways:

  1. Graphically: A line segment with an arrow. The length of the line shows the magnitude, and the arrow shows the direction.
    If a vector starts at point A and ends at point B, we write it as \(\vec{AB}\) or \(\mathbf{AB}\).
  2. Algebraically: Using a single bold lowercase letter, like \(\mathbf{a}\).
Key Takeaway: A vector describes movement or position change by stating both how far and in what direction.

1. Column Vectors in Two Dimensions (E8.2)

Since we are working in two dimensions (a flat plane), we use column vectors to describe the movement relative to the x-axis and y-axis. This is the most common way you will see vectors represented.

A column vector \(\mathbf{a}\) is written as:

$$ \mathbf{a} = \begin{pmatrix} x \\ y \end{pmatrix} $$

  • The top number (\(x\)) tells you the horizontal movement (right/left).
  • The bottom number (\(y\)) tells you the vertical movement (up/down).

Interpreting the Components

  • Positive \(x\): Movement to the right.
  • Negative \(x\): Movement to the left.
  • Positive \(y\): Movement upwards.
  • Negative \(y\): Movement downwards.

Example: The vector \(\mathbf{v} = \begin{pmatrix} 3 \\ -2 \end{pmatrix}\) represents a movement of 3 units right and 2 units down.

Connection to Transformations: Remember translations? A translation of a shape is exactly the same concept as moving the shape by a specific vector!

Quick Review: The column vector \(\begin{pmatrix} x \\ y \end{pmatrix}\) is essentially the instructions for "Go \(x\) units sideways, then \(y\) units up/down."

2. Operations with Vectors (E8.2)

A. Adding and Subtracting Vectors

Adding vectors is very straightforward. You just add the corresponding components.

Let \(\mathbf{a} = \begin{pmatrix} x_1 \\ y_1 \end{pmatrix}\) and \(\mathbf{b} = \begin{pmatrix} x_2 \\ y_2 \end{pmatrix}\).

Vector Addition

To find \(\mathbf{a} + \mathbf{b}\):

$$ \mathbf{a} + \mathbf{b} = \begin{pmatrix} x_1 \\ y_1 \end{pmatrix} + \begin{pmatrix} x_2 \\ y_2 \end{pmatrix} = \begin{pmatrix} x_1 + x_2 \\ y_1 + y_2 \end{pmatrix} $$

Example: If \(\mathbf{a} = \begin{pmatrix} 5 \\ 3 \end{pmatrix}\) and \(\mathbf{b} = \begin{pmatrix} -1 \\ 4 \end{pmatrix}\):
$$ \mathbf{a} + \mathbf{b} = \begin{pmatrix} 5 + (-1) \\ 3 + 4 \end{pmatrix} = \begin{pmatrix} 4 \\ 7 \end{pmatrix} $$

Geometrical Meaning (The Triangle Rule): If you draw vector \(\mathbf{a}\) and then draw vector \(\mathbf{b}\) starting where \(\mathbf{a}\) finished (head-to-tail), the resulting vector \(\mathbf{a} + \mathbf{b}\) is the path from the start of \(\mathbf{a}\) to the end of \(\mathbf{b}\).

Vector Subtraction

Subtraction is just adding the negative of the second vector.

$$ \mathbf{a} - \mathbf{b} = \begin{pmatrix} x_1 \\ y_1 \end{pmatrix} - \begin{pmatrix} x_2 \\ y_2 \end{pmatrix} = \begin{pmatrix} x_1 - x_2 \\ y_1 - y_2 \end{pmatrix} $$

Example: If \(\mathbf{a} = \begin{pmatrix} 5 \\ 3 \end{pmatrix}\) and \(\mathbf{b} = \begin{pmatrix} -1 \\ 4 \end{pmatrix}\):
$$ \mathbf{a} - \mathbf{b} = \begin{pmatrix} 5 - (-1) \\ 3 - 4 \end{pmatrix} = \begin{pmatrix} 6 \\ -1 \end{pmatrix} $$

🛑 Common Mistake: Don't forget your negative signs when subtracting! Subtracting a negative number turns into addition (e.g., \(5 - (-1) = 6\)).

B. Multiplying a Vector by a Scalar (Scalar Multiplication)

A scalar is just a number (like 2 or -0.5). When you multiply a vector by a scalar, you scale its length, but the direction either stays the same or reverses.

Let \(\mathbf{a} = \begin{pmatrix} x \\ y \end{pmatrix}\) and \(k\) be a scalar.

$$ k\mathbf{a} = k \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} kx \\ ky \end{pmatrix} $$

You multiply *both* the \(x\) and \(y\) components by the scalar \(k\).

  • If \(k > 0\), the vector is stretched or shrunk, but keeps the same direction.
  • If \(k < 0\), the vector points in the opposite direction (a 180° rotation).

Example 1 (Scaling): If \(\mathbf{c} = \begin{pmatrix} 4 \\ 1 \end{pmatrix}\), then \(3\mathbf{c}\) is: $$ 3\mathbf{c} = \begin{pmatrix} 3 \times 4 \\ 3 \times 1 \end{pmatrix} = \begin{pmatrix} 12 \\ 3 \end{pmatrix} $$ The new vector is three times as long, pointing in the same direction.

Example 2 (Reversing Direction): If \(\mathbf{d} = \begin{pmatrix} 2 \\ 5 \end{pmatrix}\), then \(-\mathbf{d}\) is: $$ -\mathbf{d} = -1 \begin{pmatrix} 2 \\ 5 \end{pmatrix} = \begin{pmatrix} -2 \\ -5 \end{pmatrix} $$ The new vector has the same length but points in the exact opposite direction.

Key Takeaway: Vector operations (addition, subtraction, scalar multiplication) are performed component-wise: just operate on the top numbers together and the bottom numbers together.

3. Magnitude of a Vector (E8.3)

The magnitude of a vector is simply its length or size. We don't care about the direction here, just how long the line segment is.

Notation for Magnitude

The magnitude of a vector \(\mathbf{a}\) is denoted by the modulus signs: \(|\mathbf{a}|\) or \(|\vec{AB}|\).

Calculating Magnitude using Pythagoras

Since a vector \(\begin{pmatrix} x \\ y \end{pmatrix}\) is composed of a horizontal component (\(x\)) and a vertical component (\(y\)), these two movements form the two perpendicular sides of a right-angled triangle. The vector itself is the hypotenuse!

Therefore, we can use Pythagoras’ Theorem to find the length.

For a vector \(\mathbf{a} = \begin{pmatrix} x \\ y \end{pmatrix}\):

$$ |\mathbf{a}| = \sqrt{x^2 + y^2} $$

Memory Trick: Magnitude = Length. Length in 2D = Pythagoras.

Example: Calculate the magnitude of vector \(\mathbf{v} = \begin{pmatrix} 4 \\ -3 \end{pmatrix}\).

Step 1: Identify the components: \(x=4\), \(y=-3\).

Step 2: Apply the magnitude formula:

$$ |\mathbf{v}| = \sqrt{4^2 + (-3)^2} $$

Step 3: Calculate the squares. (Remember: squaring a negative always results in a positive! \((-3)^2 = 9\)).

$$ |\mathbf{v}| = \sqrt{16 + 9} = \sqrt{25} $$

Step 4: Find the final value.

$$ |\mathbf{v}| = 5 $$

The length (magnitude) of the vector is 5 units.

Did you know? A vector with a magnitude of 1 is called a unit vector. These are very useful in higher-level mathematics for defining pure direction.

Key Takeaway: The magnitude of a vector \(\begin{pmatrix} x \\ y \end{pmatrix}\) is always found using \( \sqrt{x^2 + y^2} \). It is the distance the vector covers, regardless of the direction.

⭐ Summary of Core Vector Concepts ⭐

Vectors in 2D are a fantastic blend of geometry and algebra. To succeed in this topic, remember these four key rules:

  1. Definition: A vector has magnitude (size) and direction.
  2. Notation: We use column vectors \(\begin{pmatrix} x \\ y \end{pmatrix}\), where \(x\) is horizontal and \(y\) is vertical displacement.
  3. Operations: Add/subtract vectors by adding/subtracting the components. Multiply by a scalar by multiplying both components.
  4. Magnitude: Calculate the length using Pythagoras: \(|\mathbf{a}| = \sqrt{x^2 + y^2}\).

Keep practising those calculations, and you'll be navigating the mathematical plane like a pro!