Mathematics (0580) | Vectors in two dimensions

🚀 Understanding Vectors in Two Dimensions (0580 IGCSE Mathematics)

Hello mathematicians! Welcome to the exciting world of vectors. This topic is all about movement, position, and direction. If you ever wondered how your GPS knows where you are going, you've already touched upon vectors!

In this chapter, we will learn how to describe movement mathematically, which is super important in physics, engineering, and navigation. Don't worry if this seems tricky at first; we will break down the concepts into simple, manageable steps.

---

1. What Exactly is a Vector?

Scalar vs. Vector: The Crucial Difference

In Mathematics and Physics, we deal with two types of quantities:

• Scalar Quantity: A quantity that only has magnitude (size).
  Example: Distance (5 km), speed (60 km/h), temperature (25°C), time (2 hours).
• Vector Quantity: A quantity that has both magnitude (size) and direction.
  Example: Displacement (5 km North), velocity (60 km/h East), force (10 N downwards).

Analogy: Imagine a treasure hunt. If you are told to walk "5 metres" (Scalar), you don't know where to go. If you are told to walk "5 metres East" (Vector), you have both the distance and the direction!

Key Takeaway:

A vector is an instruction for movement: "How far?" and "In which direction?"

---

2. Representing Vectors in Two Dimensions (E8.2)

2.1 Column Vectors and Translation

In IGCSE Mathematics, we represent two-dimensional vectors using column vectors. These are written like stacked numbers, describing the movement relative to the \(x\) and \(y\) axes.

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

\[ \mathbf{a} = \begin{pmatrix} x \\ y \end{pmatrix} \]

• The top number, \(x\), describes the horizontal movement:
  • Positive \(x\) means movement to the Right.
  • Negative \(x\) means movement to the Left.
• The bottom number, \(y\), describes the vertical movement:
  • Positive \(y\) means movement Up.
  • Negative \(y\) means movement Down.

Connection to Translations: When you translate a shape on a graph (C8.1.4), you are moving it by exactly one vector!

Example: The vector \( \begin{pmatrix} 4 \\ -2 \end{pmatrix} \) means move 4 units right and 2 units down.

2.2 Vector Notation

Vectors can be denoted in three main ways (as referenced in the syllabus notes):

1. Bold, lowercase letter: \(\mathbf{a}\) or \(\mathbf{b}\).
2. Vector between two points: \(\mathbf{AB}\) (The vector from point A to point B).
3. Column Vector: \( \begin{pmatrix} x \\ y \end{pmatrix} \).

Did you know? In handwritten work, since you can't easily write in bold, students often draw a little arrow above the letter, like \(\vec{a}\).

If the vector starts at the origin (0, 0) and ends at point P\((4, 3)\), it is called the position vector of P, written as \(\mathbf{p}\) or \(\mathbf{OP}\).

---

3. Vector Operations (E8.2)

Working with vectors algebraically is very straightforward because you simply treat the \(x\) and \(y\) components separately.

3.1 Adding Vectors (Vector Addition)

To add two vectors, you add the corresponding components.

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

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

This is the "head-to-tail" rule graphically. If you follow the movement instruction for \(\mathbf{a}\) and then follow the movement instruction for \(\mathbf{b}\), the resultant vector \( \mathbf{a} + \mathbf{b} \) is the single direct path from the starting point to the final endpoint.

Example 1: Find the resultant vector \(\mathbf{r} = \mathbf{p} + \mathbf{q}\) where \( \mathbf{p} = \begin{pmatrix} 3 \\ 5 \end{pmatrix} \) and \( \mathbf{q} = \begin{pmatrix} 1 \\ -2 \end{pmatrix} \).

\[ \mathbf{r} = \begin{pmatrix} 3 + 1 \\ 5 + (-2) \end{pmatrix} = \begin{pmatrix} 4 \\ 3 \end{pmatrix} \]

3.2 Subtracting Vectors (Vector Subtraction)

Subtracting a vector is the same as adding its negative vector. The negative vector, \(-\mathbf{b}\), has the same magnitude but points in the exact opposite direction.

If \( \mathbf{b} = \begin{pmatrix} x_2 \\ y_2 \end{pmatrix} \), then \( -\mathbf{b} = \begin{pmatrix} -x_2 \\ -y_2 \end{pmatrix} \).

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

Example 2: Find \(\mathbf{c} - \mathbf{d}\) where \( \mathbf{c} = \begin{pmatrix} 8 \\ 1 \end{pmatrix} \) and \( \mathbf{d} = \begin{pmatrix} -4 \\ 6 \end{pmatrix} \).

\[ \mathbf{c} - \mathbf{d} = \begin{pmatrix} 8 - (-4) \\ 1 - 6 \end{pmatrix} = \begin{pmatrix} 8 + 4 \\ -5 \end{pmatrix} = \begin{pmatrix} 12 \\ -5 \end{pmatrix} \]

3.3 Multiplying by a Scalar

Multiplying a vector by a normal number (called a scalar, \(k\)) changes its magnitude (length) but not its direction (unless \(k\) is negative). You multiply both components by the scalar.

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

• If \(k > 0\), the new vector is parallel and longer (if \(k > 1\)) or shorter (if \(0 < k < 1\)).
• If \(k < 0\), the new vector is parallel but points in the opposite direction.

Example 3: If \( \mathbf{m} = \begin{pmatrix} 2 \\ -3 \end{pmatrix} \), find \( 4\mathbf{m} \) and \( -2\mathbf{m} \).

\[ 4\mathbf{m} = 4 \begin{pmatrix} 2 \\ -3 \end{pmatrix} = \begin{pmatrix} 4 \times 2 \\ 4 \times (-3) \end{pmatrix} = \begin{pmatrix} 8 \\ -12 \end{pmatrix} \]

\[ -2\mathbf{m} = -2 \begin{pmatrix} 2 \\ -3 \end{pmatrix} = \begin{pmatrix} -2 \times 2 \\ -2 \times (-3) \end{pmatrix} = \begin{pmatrix} -4 \\ 6 \end{pmatrix} \]

Common Mistake to Avoid:

When multiplying by a negative scalar (like \(-2\)), make sure you multiply both the top and bottom numbers by \(-2\). The double negative in the second component becomes a positive!

Key Takeaway:

Vector arithmetic is just simple arithmetic applied to the \(x\) and \(y\) components separately.

---

4. The Magnitude of a Vector (E8.3)

The magnitude of a vector is simply its length. Since a vector describes movement in two dimensions (a horizontal change and a vertical change), we can always find its length using the most famous theorem in geometry: Pythagoras' theorem!

4.1 Formula for Magnitude

If a vector \(\mathbf{a} = \begin{pmatrix} x \\ y \end{pmatrix}\), the magnitude (or length) is denoted by \(|\mathbf{a}|\) or \(| \mathbf{AB} |\) (using modulus signs).

Imagine \(x\) and \(y\) as the two shorter sides of a right-angled triangle, and the vector itself is the hypotenuse.

The formula is:

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

Memory Aid: Magnitude is always positive (length cannot be negative!), so you must take the square root of the sum of the squares of the components.

4.2 Step-by-Step Calculation

Find the magnitude of the vector \(\mathbf{v} = \begin{pmatrix} -5 \\ 12 \end{pmatrix}\).

Step 1: Identify the components.
\(x = -5\), \(y = 12\).

Step 2: Substitute into the Pythagoras formula.
\[ |\mathbf{v}| = \sqrt{(-5)^2 + (12)^2} \]

Step 3: Calculate the squares (remember \((-5)^2\) is positive 25!).
\[ |\mathbf{v}| = \sqrt{25 + 144} \]

Step 4: Sum the components and find the square root.
\[ |\mathbf{v}| = \sqrt{169} \]
\[ |\mathbf{v}| = 13 \]

The magnitude of vector \(\mathbf{v}\) is 13 units.

4.3 Magnitude of a Vector Between Two Points

If you are given two points A\((x_A, y_A)\) and B\((x_B, y_B)\), first you must find the vector \(\mathbf{AB}\).

\[ \mathbf{AB} = \mathbf{b} - \mathbf{a} = \begin{pmatrix} x_B - x_A \\ y_B - y_A \end{pmatrix} \]

Then you find the magnitude of this resultant vector.

Example 4: Find the length of the line segment joining P(1, 4) and Q(4, 8).

1. Find the vector \(\mathbf{PQ}\):
\[ \mathbf{PQ} = \begin{pmatrix} 4 - 1 \\ 8 - 4 \end{pmatrix} = \begin{pmatrix} 3 \\ 4 \end{pmatrix} \]

2. Find the magnitude \(|\mathbf{PQ}|\):
\[ |\mathbf{PQ}| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

Key Takeaway:

The magnitude of a vector is simply the distance between its start and end point, calculated using Pythagoras.

---

Quick Vector Toolkit Review

Keep practising these calculations. Once you master vector arithmetic and magnitude, you'll find that many vector problems are just simple extensions of coordinate geometry! You've got this!