Mathematics (0580) | Vector geometry

Welcome to Vector Geometry!

Hello! Vectors might look a little complicated with all the arrows and strange brackets, but don't worry. This chapter is about learning how to describe movement accurately. Think of vectors as directions on a treasure map: they tell you exactly how far to walk and in what direction.

In IGCSE Mathematics (0580) Extended syllabus, we use vectors in two dimensions (flat space) to describe translations, paths, and positions, which is super useful in physics and engineering. Let's break it down!

Section 1: What Exactly Is a Vector?

1.1 Scalars vs. Vectors: The Key Difference

In mathematics and physics, quantities are classified based on whether they involve direction.

• Scalar Quantity: A quantity that only has magnitude (size).
  Example: Speed (50 km/h), Mass (10 kg), Time (2 hours).
• Vector Quantity: A quantity that has both magnitude (size) and direction.
  Example: Velocity (50 km/h North), Force (10 N downwards), Displacement (2 km East).

Analogy: If you say you ran 5 km, that's a scalar (distance). If you say you ran 5 km towards the school, that's a vector (displacement).

1.2 Vector Notation

We use specific notation to show that a quantity is a vector:

1. Bold Letter: In textbooks or print, vectors are usually written in bold lowercase letters, like a or b.
2. Arrow Notation: When handwritten, you draw a line and an arrow above the letter, like \(\vec{a}\).
3. Points Notation: If a vector goes from point A to point B, we write it as \(\vec{AB}\).

Section 2: Column Vectors

In IGCSE Math, we usually work with vectors using a useful format called the column vector, especially when describing translations.

2.1 Defining a Column Vector

A column vector is written like this: $$ \mathbf{a} = \begin{pmatrix} x \\ y \end{pmatrix} $$ Where:

• \(x\) represents the horizontal movement. (Right is positive, left is negative).
• \(y\) represents the vertical movement. (Up is positive, down is negative).

Example: The vector \(\begin{pmatrix} 3 \\ -5 \end{pmatrix}\) means move 3 units right and 5 units down.
Example: If a translation moves a point 2 units left and 1 unit up, the vector is \(\begin{pmatrix} -2 \\ 1 \end{pmatrix}\).

2.2 Finding a Vector Between Two Points

If you have the coordinates of two points, \(A(x_1, y_1)\) and \(B(x_2, y_2)\), the vector \(\vec{AB}\) is found by subtracting the start coordinates from the end coordinates (End minus Start).

$$ \vec{AB} = \begin{pmatrix} x_2 - x_1 \\ y_2 - y_1 \end{pmatrix} $$

Step-by-Step Example:

Find the vector \(\vec{PQ}\) if \(P\) is \((1, 8)\) and \(Q\) is \((6, 5)\).

1. Identify the coordinates: \(P(1, 8)\) and \(Q(6, 5)\).
2. Calculate the horizontal change (\(x\)): \(6 - 1 = 5\).
3. Calculate the vertical change (\(y\)): \(5 - 8 = -3\).
4. Write the column vector: $$ \vec{PQ} = \begin{pmatrix} 5 \\ -3 \end{pmatrix} $$

Common Mistake Alert! Always remember: a column vector describes movement, not a position. The notation \(\begin{pmatrix} 3 \\ 5 \end{pmatrix}\) is a vector; the notation \((3, 5)\) is a coordinate point. They are similar but represent different things!

Section 3: Operations with Vectors

We can add, subtract, and multiply vectors by a scalar (a simple number).

3.1 Vector Addition (Adding Journeys)

If you make one journey (\(\mathbf{a}\)) and then another journey (\(\mathbf{b}\)), the resulting single journey is \(\mathbf{a} + \mathbf{b}\). This is called the Triangle Rule.
If we have \(\mathbf{a} = \begin{pmatrix} x_1 \\ y_1 \end{pmatrix}\) and \(\mathbf{b} = \begin{pmatrix} x_2 \\ y_2 \end{pmatrix}\): $$ \mathbf{a} + \mathbf{b} = \begin{pmatrix} x_1 + x_2 \\ y_1 + y_2 \end{pmatrix} $$

Example: Adding Vectors

If \(\mathbf{p} = \begin{pmatrix} 2 \\ 5 \end{pmatrix}\) and \(\mathbf{q} = \begin{pmatrix} 1 \\ -3 \end{pmatrix}\):
$$ \mathbf{p} + \mathbf{q} = \begin{pmatrix} 2 + 1 \\ 5 + (-3) \end{pmatrix} = \begin{pmatrix} 3 \\ 2 \end{pmatrix} $$

3.2 Vector Subtraction (Reversing a Journey)

Subtracting a vector is the same as adding its negative.
The negative vector, \(-\mathbf{a}\), simply reverses the direction of \(\mathbf{a}\).

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

Example: Subtracting Vectors

Using the vectors \(\mathbf{p}\) and \(\mathbf{q}\) above: $$ \mathbf{p} - \mathbf{q} = \begin{pmatrix} 2 - 1 \\ 5 - (-3) \end{pmatrix} = \begin{pmatrix} 1 \\ 8 \end{pmatrix} $$

3.3 Scalar Multiplication (Scaling)

Multiplying a vector by a scalar (a number, let's call it \(k\)) changes its length, but not its direction (unless \(k\) is negative). You multiply both the \(x\) and \(y\) components by the scalar \(k\).

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

Example: Scaling

If \(\mathbf{v} = \begin{pmatrix} 4 \\ -1 \end{pmatrix}\), find \(3\mathbf{v}\) and \(-2\mathbf{v}\).

$$ 3\mathbf{v} = 3 \begin{pmatrix} 4 \\ -1 \end{pmatrix} = \begin{pmatrix} 3 \times 4 \\ 3 \times (-1) \end{pmatrix} = \begin{pmatrix} 12 \\ -3 \end{pmatrix} $$ $$ -2\mathbf{v} = -2 \begin{pmatrix} 4 \\ -1 \end{pmatrix} = \begin{pmatrix} -2 \times 4 \\ -2 \times (-1) \end{pmatrix} = \begin{pmatrix} -8 \\ 2 \end{pmatrix} $$ Note how \(-2\mathbf{v}\) points in the opposite direction (because of the negative sign) and is twice as long (because the magnitude is doubled).

3.4 Parallel Vectors

Two vectors are parallel if one is a scalar multiple of the other.
If \(\mathbf{a} = k\mathbf{b}\), then \(\mathbf{a}\) is parallel to \(\mathbf{b}\).

Example: \(\mathbf{r} = \begin{pmatrix} 6 \\ 9 \end{pmatrix}\) and \(\mathbf{s} = \begin{pmatrix} 2 \\ 3 \end{pmatrix}\). Since \(\mathbf{r} = 3\mathbf{s}\), the vectors \(\mathbf{r}\) and \(\mathbf{s}\) are parallel.

Section 4: The Magnitude of a Vector

The magnitude of a vector is simply its length or size. We calculate this using Pythagoras' theorem, because the \(x\) and \(y\) movements form the two perpendicular sides of a right-angled triangle, and the vector itself is the hypotenuse.

4.1 Magnitude Notation and Formula (E8.3)

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

For a vector \(\mathbf{a} = \begin{pmatrix} x \\ y \end{pmatrix}\), the magnitude is: $$ |\mathbf{a}| = \sqrt{x^2 + y^2} $$

Step-by-Step Example: Calculating Magnitude

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

1. Square the components: \(x^2 = 5^2 = 25\), and \(y^2 = (-12)^2 = 144\).
2. Add them together: \(25 + 144 = 169\).
3. Take the square root: $$ |\mathbf{v}| = \sqrt{169} = 13 $$ The length of the vector is 13 units.

4.2 Distance Between Two Points Using Magnitude (E4.3, reinforced by E8.3)

Finding the distance between two points \(A\) and \(B\) is exactly the same as finding the magnitude of the vector \(\vec{AB}\).

If \(A=(x_1, y_1)\) and \(B=(x_2, y_2)\), the distance \(d\) is: $$ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} $$ This is just the magnitude formula applied to the components of \(\vec{AB}\)! Don't worry if this seems tricky at first—it's just Pythagoras in disguise!

Section 5: Applying Vectors in Geometry

Vectors are fantastic tools for solving geometry problems, especially those involving parallel lines, midpoints, and straight lines.

5.1 Finding a Path (Vector Routes)

In complex diagrams (like parallelograms or triangles), you often need to find a vector representing a path between two points that isn't directly drawn. You do this by combining known vectors.

Rule: To go from P to Q, find a sequence of vectors that takes you from P to Q.

If you want to find \(\vec{AC}\) in a parallelogram \(ABCD\): $$ \vec{AC} = \vec{AB} + \vec{BC} $$

Example: Using Negative Vectors

If \(\vec{OA} = \mathbf{a}\) and \(\vec{OB} = \mathbf{b}\). Find the vector \(\vec{AB}\).

1. To get from A to B, we cannot go directly (unless it is given).
2. We must go backward along \(\vec{OA}\) (which is \(\vec{AO}\)), and then forward along \(\vec{OB}\).
3. The vector \(\vec{AO}\) is \(-\mathbf{a}\).
4. Therefore: $$ \vec{AB} = \vec{AO} + \vec{OB} = -\mathbf{a} + \mathbf{b} \quad \text{or} \quad \mathbf{b} - \mathbf{a} $$

5.2 Midpoints and Ratios in Vectors

Vectors help us describe points lying along a line segment.

• If \(M\) is the midpoint of the line segment \(AB\): $$ \vec{AM} = \frac{1}{2} \vec{AB} $$
• If point \(P\) divides \(AB\) in the ratio 1:3, then \(P\) is 1/4 of the way along \(AB\): $$ \vec{AP} = \frac{1}{4} \vec{AB} $$

Example: Combining Vectors and Ratios

In a triangle \(OAB\), \(\vec{OA} = \mathbf{a}\) and \(\vec{OB} = \mathbf{b}\). Point \(M\) is the midpoint of \(AB\). Find \(\vec{OM}\) in terms of \(\mathbf{a}\) and \(\mathbf{b}\).

1. First, find the vector representing the full side \(\vec{AB}\): $$ \vec{AB} = \mathbf{b} - \mathbf{a} $$ 2. Since \(M\) is the midpoint, \(\vec{AM} = \frac{1}{2} \vec{AB}\): $$ \vec{AM} = \frac{1}{2}(\mathbf{b} - \mathbf{a}) $$ 3. Now, find the path \(\vec{OM}\) by going from \(O\) to \(A\), and then \(A\) to \(M\): $$ \vec{OM} = \vec{OA} + \vec{AM} = \mathbf{a} + \frac{1}{2}(\mathbf{b} - \mathbf{a}) $$ 4. Simplify the expression: $$ \vec{OM} = \mathbf{a} + \frac{1}{2}\mathbf{b} - \frac{1}{2}\mathbf{a} = \frac{1}{2}\mathbf{a} + \frac{1}{2}\mathbf{b} $$

5.3 Testing for Collinearity (Points on a Straight Line)

Three points A, B, and C are collinear (lie on the same straight line) if the vectors between them are parallel and share a common point.

If \(\vec{AB} = k \cdot \vec{BC}\), and they share point B, then A, B, and C are collinear.