Vectors in 2D/3D (HL deeper) - Mathematics: Analysis & Approaches IB Study Notes

Think of a vector like a treasure map instruction that says: 'Go 5 steps North-East.' It tells you both which way to go (North-East) and how far to go (5 steps). It's different from just a number (like '5 steps') because that doesn't tell you the direction.

• 2D (Two Dimensions): This is like drawing on a flat piece of paper or a computer screen. You can go left/right (x-axis) and up/down (y-axis). A vector here might look like an arrow on a map.
• 3D (Three Dimensions): This is like the real world! You can go left/right (x-axis), up/down (y-axis), AND forward/backward (z-axis, for height or depth). Imagine a bird flying – it moves in 3D. A vector in 3D tells you how to move in all three directions.

We represent vectors using special notation, usually a letter with an arrow above it (like $\vec{a}$) or in bold ($\mathbf{a}$). We also write them as a column of numbers, like $\begin{pmatrix} 3 \ 4 \end{pmatrix}$ for 2D, meaning 'go 3 units right and 4 units up', or $\begin{pmatrix} 1 \ 2 \ 5 \end{pmatrix}$ for 3D, meaning 'go 1 unit right, 2 units up, and 5 units forward'.

Deeper Dive for HL: We're not just looking at simple movements. We'll explore how to find the angle between two vectors (imagine two roads meeting, what's the angle?), how to check if they are parallel (running in the same direction, like train tracks) or perpendicular (meeting at a perfect right angle, like the corner of a room), and even how to project one vector onto another (like finding the shadow of a stick on the ground).

Let's imagine you're a drone pilot, and you need to program your drone to fly a specific path.

1. Starting Point: Your drone is at a specific location, let's say the origin of our coordinate system (0,0,0) – like the center of a giant invisible grid.
2. First Flight Segment: You want the drone to fly 10 meters East, 5 meters North, and 2 meters Up. We can represent this as a vector: $\vec{a} = \begin{pmatrix} 10 \ 5 \ 2 \end{pmatrix}$. This vector tells the drone exactly how to move from its starting point.
3. Second Flight Segment: From its new position, you want the drone to fly another 3 meters West (so -3 East), 8 meters North, and then land 7 meters Down (so -7 Up). This is another vector: $\vec{b} = \begin{pmatrix} -3 \ 8 \ -7 \end{pmatrix}$.
4. Total Displacement: To find out where the drone ends up from its original starting point, you simply add these vectors together! $\vec{a} + \vec{b} = \begin{pmatrix} 10 \ 5 \ 2 \end{pmatrix} + \begin{pmatrix} -3 \ 8 \ -7 \end{pmatrix} = \begin{pmatrix} 10 + (-3) \ 5 + 8 \ 2 + (-7) \end{pmatrix} = \begin{pmatrix} 7 \ 13 \ -5 \end{pmatrix}$.

This final vector, $\begin{pmatrix} 7 \ 13 \ -5 \end{pmatrix}$, tells you the drone's final position relative to its starting point: 7 meters East, 13 meters North, and 5 meters Down. This is its resultant vector (the single vector that represents the combined effect of multiple vectors). Drone pilots, game developers, and even astronauts use vectors like this all the time!