Welcome to the World of Position Vectors!
In your previous maths studies, you’ve used coordinates like \( (3, 4) \) to mark a spot on a graph. In A Level Mathematics B (MEI), we take this a step further by using Position Vectors. Think of a position vector as a set of instructions that tells you exactly how to get from a fixed "Home" base (the Origin) to any specific point in space. It’s like giving someone a GPS coordinate, but expressing it as a movement.
Don't worry if vectors felt a bit abstract before—position vectors are very grounded because they are always "anchored" to the same starting point!
1. What exactly is a Position Vector?
Most vectors are "free"—you can slide them around the graph as long as they point in the same direction and have the same length. However, a Position Vector is special. It is a vector that starts at the Origin \( O (0, 0) \) and ends at a specific point, let's call it \( A \).
Key Notation
- The position vector of point \( A \) is written as \( \vec{OA} \).
- In textbook print, it is often represented by a bold lowercase letter, like a.
- In your own handwriting, you should write it as \( \underline{a} \) or with an arrow \( \vec{a} \).
The Connection to Coordinates
If a point \( A \) has coordinates \( (x, y) \), its position vector is simply:
\( \mathbf{a} = \begin{pmatrix} x \\ y \end{pmatrix} \) or \( \mathbf{a} = x\mathbf{i} + y\mathbf{j} \)
Example: If point P is at \( (5, -2) \), then its position vector \( \vec{OP} \) is \( \begin{pmatrix} 5 \\ -2 \end{pmatrix} \).
Quick Review: The Origin Is Key
Always remember: A position vector must start at the origin \( O \). If it starts anywhere else, it's just a general vector (often called a displacement vector).
Takeaway: Position vectors are just a "vector version" of coordinates. They represent the journey from \( (0,0) \) to a point.
2. Finding the Vector Between Two Points
This is one of the most important skills in this chapter. If you know the position of point \( A \) (vector a) and the position of point \( B \) (vector b), how do you find the vector that goes directly from \( A \) to \( B \)?
The "Destination Minus Start" Rule
To find the vector \( \vec{AB} \), you use this simple formula:
\( \vec{AB} = \mathbf{b} - \mathbf{a} \)
Why does this work?
Think of it as a detour. To get from \( A \) to \( B \), you could go backwards from \( A \) to the Origin (\( -\mathbf{a} \)) and then from the Origin to \( B \) (\( +\mathbf{b} \)).
So, \( \vec{AB} = \vec{AO} + \vec{OB} = -\mathbf{a} + \mathbf{b} \), which we usually write as \( \mathbf{b} - \mathbf{a} \).
Memory Aid: "B - A"
To find the vector \( \vec{AB} \), just remember: End minus Start.
(The 2nd letter's vector minus the 1st letter's vector).
Example:
If \( \mathbf{a} = \begin{pmatrix} 2 \\ 3 \end{pmatrix} \) and \( \mathbf{b} = \begin{pmatrix} 10 \\ 5 \end{pmatrix} \), then:
\( \vec{AB} = \begin{pmatrix} 10 \\ 5 \end{pmatrix} - \begin{pmatrix} 2 \\ 3 \end{pmatrix} = \begin{pmatrix} 8 \\ 2 \end{pmatrix} \).
Takeaway: You can find the vector between any two points by subtracting the "start" position vector from the "finish" position vector.
3. Calculating Distance Between Two Points
Once you have the vector \( \vec{AB} \), you might be asked to find the distance between point \( A \) and point \( B \). This is simply the magnitude (length) of the vector \( \vec{AB} \).
Step-by-Step Process:
- Find the vector \( \vec{AB} \) using \( \mathbf{b} - \mathbf{a} \). Let's say the result is \( \begin{pmatrix} x \\ y \end{pmatrix} \).
- Use Pythagoras' Theorem to find the length:
\( \text{Distance} = |\vec{AB}| = \sqrt{x^2 + y^2} \)
Analogy: If you walk 3 blocks East and 4 blocks North, Pythagoras tells you the "as the crow flies" distance is 5 blocks.
Common Mistake to Avoid
Don't forget to square the numbers before adding them! Also, if a component is negative, like \( -3 \), squaring it makes it positive: \( (-3)^2 = 9 \). Distances are never negative!
Takeaway: Distance is just the magnitude of the vector connecting two points. Use \( \sqrt{x^2 + y^2} \).
4. Moving into Three Dimensions (3D)
The MEI syllabus (Mv7) requires you to apply these same rules to 3D space. The only difference is that we add a third component, \( z \), and a third unit vector, k.
- Position Vector in 3D: \( \mathbf{r} = \begin{pmatrix} x \\ y \\ z \end{pmatrix} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k} \)
- Subtraction: It still works exactly the same way! \( \vec{AB} = \mathbf{b} - \mathbf{a} \), just with three numbers to subtract instead of two.
- 3D Distance: \( |\vec{AB}| = \sqrt{x^2 + y^2 + z^2} \)
Did you know?
3D vectors are used by airplane pilots and air traffic controllers every day to describe a plane's position relative to the airport (the origin), including its altitude (the \( z \) component).
Takeaway: 3D vectors look more complex, but the math is identical to 2D. Just add the extra \( z \) step!
Summary Quick-Check
Before you move on to practice questions, make sure you are comfortable with these "Must-Knows":
- Position Vector: A vector starting from the origin \( O \).
- \( \vec{AB} = \mathbf{b} - \mathbf{a} \): The golden rule for finding the vector between points.
- Magnitude: Use \( \sqrt{x^2 + y^2 (+ z^2)} \) to find the distance between points.
- Coordinates: The numbers in the coordinate \( (x, y) \) are the same as the components in the position vector \( \begin{pmatrix} x \\ y \end{pmatrix} \).
Final Encouragement: Position vectors are the bridge between geometry and algebra. Once you master the "End minus Start" rule, you've unlocked the hardest part of this chapter! Keep practicing, and it will become second nature.