Introduction: How Far Apart Are We?
Welcome to one of the most practical parts of the Pure Mathematics: Vectors section! In this chapter, we are going to learn how to find the exact distance between two points in space using vectors.
Whether you are designing a level in a video game, plotting a flight path, or just trying to find the shortest route to a coffee shop, you are using the distance between points. If you’ve ever used Pythagoras’ Theorem, you’re already halfway there. Don’t worry if vectors feel a bit strange right now—we’ll break it down step-by-step!
The Concept: Pythagoras in Disguise
When we talk about the distance between two points, \(A\) and \(B\), we are really looking for the magnitude (the length) of the vector that connects them.
Imagine point \(A\) is at your house and point \(B\) is at the park. To find the straight-line distance, we look at how far across (\(i\) direction) and how far up (\(j\) direction) we need to travel to get from \(A\) to \(B\). This creates a right-angled triangle!
The 2D Formula
According to the OCR syllabus (Ref 1.10f), if you have two points represented by position vectors:
Point \(A = a\mathbf{i} + b\mathbf{j}\)
Point \(B = c\mathbf{i} + d\mathbf{j}\)
The distance between them is:
Distance \(= \sqrt{(c - a)^2 + (d - b)^2}\)
Analogy: Think of \((c - a)\) as the "horizontal gap" and \((d - b)\) as the "vertical gap." You square them, add them, and square root the total—just like finding the hypotenuse in GCSE geometry!
Step-by-Step: How to Calculate Distance
Follow these steps to ensure you never get lost in the algebra:
- Identify the components: Write down the \(i\) and \(j\) values for both points.
- Find the differences: Subtract the components of the first vector from the second. (It doesn't actually matter which way you subtract because squaring makes everything positive anyway!)
- Square the results: Square both of those differences.
- Sum them up: Add those two squares together.
- Square root: Take the square root of your final sum.
Example: Find the distance between position vectors \(\mathbf{a} = 2\mathbf{i} + 5\mathbf{j}\) and \(\mathbf{b} = 6\mathbf{i} + 2\mathbf{j}\).
1. Gap in \(i\): \(6 - 2 = 4\)
2. Gap in \(j\): \(2 - 5 = -3\)
3. Square them: \(4^2 = 16\) and \((-3)^2 = 9\)
4. Add: \(16 + 9 = 25\)
5. Root: \(\sqrt{25} = 5\)
The distance is 5 units.
Stepping into 3D
The H240 syllabus (Ref 1.10b) also requires you to work in three dimensions (\(\mathbf{i}, \mathbf{j}, \mathbf{k}\)). The great news? The logic is exactly the same! You just add one more "gap" for the \(k\) component.
For points \((x_1\mathbf{i} + y_1\mathbf{j} + z_1\mathbf{k})\) and \((x_2\mathbf{i} + y_2\mathbf{j} + z_2\mathbf{k})\):
Distance \(= \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}\)
Quick Review Box
Distance = \(\sqrt{(\Delta i)^2 + (\Delta j)^2 + (\Delta k)^2}\)
(Where \(\Delta\) just means "the difference in")
Common Mistakes to Avoid
Even the best mathematicians trip up on these simple things. Keep an eye out for:
- The "Negative Trap": When you square a negative number, like \((-4)^2\), the answer is always positive (16, not -16). If you get a math error on your calculator, check this first!
- Mixing up components: Make sure you subtract \(i\) from \(i\) and \(j\) from \(j\). Don't cross the streams!
- Forgetting the Square Root: It’s easy to do all the hard work and forget the very last step. Always check if your answer seems reasonably sized compared to the points.
Did You Know?
Did you know? This formula is the reason your GPS works! Satellites calculate the "distance between points" (the satellite and your phone) using these 3D vector coordinates to pinpoint exactly where you are on Earth.
Summary & Key Takeaways
Key Takeaway 1:
The distance between two points in vector form is simply the magnitude of the displacement vector between them.
Key Takeaway 2:
The formula is based entirely on Pythagoras' Theorem. If you can find the length of a triangle side, you can do this!
Key Takeaway 3:
For 3D vectors, don't panic—just add the third component (\(k\)) into the square root formula exactly like the others.
Don't worry if this seems a bit abstract at first. Once you practice three or four examples, the pattern becomes second nature. You've got this!