Introduction to Vectors and Matrices in Motion

Welcome to one of the most exciting parts of the Higher Level (HL) syllabus! In this chapter, we are going to look at how mathematics describes movement and change. Think of vectors as a GPS system: they don't just tell you where you are, but also which way you are heading and how fast you are going. We will then combine this with matrices, which act like "filters" or "commands" that can flip, rotate, or stretch shapes in space. Whether you are interested in computer graphics, engineering, or physics, these tools are your bread and butter.

1. Vector Equations of a Line (AHL 3.11)

In your earlier studies, you used \(y = mx + c\) to describe a line. However, in three dimensions (3D), that formula doesn't work well. Instead, we use the vector equation of a line. It works like giving someone directions: "Start at this point, and then walk in this direction for as long as you like."

The standard form is:
\(r = a + \lambda b\)

Where:

  • \(r\) is the position vector of any point on the line \(\begin{pmatrix} x \\ y \\ z \end{pmatrix}\).
  • \(a\) is a known position vector (a fixed point on the line).
  • \(b\) is the direction vector (the way the line is pointing).
  • \(\lambda\) (lambda) is a scalar parameter. It’s a number that tells you how far to go along the direction \(b\).

Analogy: Imagine a bus route. The bus station is your vector \(a\). The direction the bus travels is vector \(b\). Depending on how much time you spend on the bus (\(\lambda\)), you will end up at a different point (\(r\)) along that route.

Quick Tip: Parallel Lines

Two lines are parallel if their direction vectors (\(b_1\) and \(b_2\)) are multiples of each other. For example, \(\begin{pmatrix} 1 \\ 2 \end{pmatrix}\) is parallel to \(\begin{pmatrix} 3 \\ 6 \end{pmatrix}\).

2. Vector Kinematics (AHL 3.12)

Kinematics is the study of motion. In this course, we use vectors to describe objects moving in 2D and 3D space. This is basically the "real-world" version of the line equation we just learned.

Constant Velocity

If an object moves at a constant velocity, its position at any time \(t\) is given by:
\(r = r_0 + vt\)

Notice the similarity?

  • \(r_0\) is the initial position (where it started at \(t = 0\)).
  • \(v\) is the velocity vector.
  • \(t\) is time (this replaces our parameter \(\lambda\)).

Important Distinction: Velocity vs. Speed
Velocity is a vector (e.g., \(\begin{pmatrix} 3 \\ 4 \end{pmatrix}\)).
Speed is a scalar and is the magnitude of the velocity vector: \(|v| = \sqrt{v_x^2 + v_y^2}\).
Example: If velocity is \(\begin{pmatrix} 3 \\ 4 \end{pmatrix}\) m/s, the speed is \(\sqrt{3^2 + 4^2} = 5\) m/s.

Variable Velocity (2D Only)

Sometimes the velocity isn't constant; it might change over time (like a car accelerating). In this case, the velocity vector \(v\) is a function of time, \(v(t)\). You might be asked to find the position at a specific time or describe the path of the object.

Key Takeaway:

For any kinematics problem, always identify your "starting point" and your "velocity vector" first. If the problem asks for the distance between two moving objects, you are looking for the magnitude of the vector connecting them (\(r_1 - r_2\)).

3. Matrix Transformations (AHL 3.9)

Matrices are powerful tools for moving points around a 2D plane. We can represent a point \((x, y)\) as a column vector \(\begin{pmatrix} x \\ y \end{pmatrix}\). When we multiply this by a \(2 \times 2\) matrix, the point "transforms" to a new location \(\begin{pmatrix} x' \\ y' \end{pmatrix}\).

Common 2D Transformations

  • Reflections:
    • In the x-axis: \(\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}\)
    • In the y-axis: \(\begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}\)
    • In the line \(y = x\): \(\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\)
  • Rotations (around origin): To rotate an object by angle \(\theta\) counter-clockwise:
    \(\begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}\)
  • Stretches/Enlargements:
    • Horizontal stretch (factor \(k\)): \(\begin{pmatrix} k & 0 \\ 0 & 1 \end{pmatrix}\)
    • Vertical stretch (factor \(k\)): \(\begin{pmatrix} 1 & 0 \\ 0 & k \end{pmatrix}\)
    • Enlargement (factor \(k\)): \(\begin{pmatrix} k & 0 \\ 0 & k \end{pmatrix}\)
  • Translations: Unlike the others, translations are usually done by adding a vector: \(\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} x \\ y \end{pmatrix} + \begin{pmatrix} a \\ b \end{pmatrix}\).

Compositions (Combining Transformations)

If you want to rotate a point and then reflect it, you multiply the matrices together.
Watch out! The order matters. If transformation \(A\) happens first and then \(B\), the math is written as \(B \times A \times \text{point}\). We work from right to left.

The Geometric Meaning of the Determinant

The determinant of a transformation matrix (\(ad - bc\)) tells us how the area of a shape changes.

  • If \(\text{det}(M) = 3\), the new shape has 3 times the area of the original.
  • If \(\text{det}(M) = 1\), the area stays the same (like in a rotation or reflection).
  • If \(\text{det}(M)\) is negative, the orientation has been reversed (like a reflection).

Summary Checklist

Did you get the basics?

  • Can you write the equation of a line given a point and a direction?
  • Can you calculate the speed of an object from its velocity vector?
  • Do you know which matrix to use for a rotation or reflection?
  • Can you use your GDC to multiply matrices for composite transformations?

Don't worry if matrices feel abstract at first. Just remember: every matrix is just a set of instructions for where the points \(\begin{pmatrix} 1 \\ 0 \end{pmatrix}\) and \(\begin{pmatrix} 0 \\ 1 \end{pmatrix}\) should move to!

Quick Review: For any line \(r = a + \lambda b\), the vector \(a\) is a position, and \(b\) is a direction. In kinematics, \(b\) represents velocity.

For more on basic vector operations like the scalar product, see the chapter "Vectors, scalar and vector products (HL)".