Welcome to the World of Position Vectors!
In your study of vectors so far, you have likely looked at "displacement vectors" — arrows that tell you how to get from one place to another (like "walk 3 miles North"). In this chapter, we focus on Position Vectors. These are special because they tell us exactly where a point is located on a map or grid. Think of it like a GPS coordinate for your math problems!
By the end of these notes, you’ll be able to identify position vectors, use them to find the distance between points, and calculate the path between any two locations. Don't worry if it feels like a lot of new notation; we will take it one step at a time.
1. What is a Position Vector?
A Position Vector is a vector that starts at the Origin (the point \( (0,0) \), usually called \( O \)) and ends at a specific point in space.
The Core Idea:
If you have a point \( P \) with coordinates \( (x, y) \), its position vector is the journey from the origin \( O \) to \( P \). We write this as \(\vec{OP}\) or simply as a bold lowercase letter p.
Notation:
In your exams, you will see position vectors written as column vectors:
If point \( P = (3, 4) \), then the position vector \(\mathbf{p} = \begin{pmatrix} 3 \\ 4 \end{pmatrix}\).
This can also be written using unit vectors: \(\mathbf{p} = 3\mathbf{i} + 4\mathbf{j}\).
Did you know?
In professional navigation, position vectors are the backbone of how airplanes and ships track their exact location relative to a fixed starting point!
Quick Review:
• The point is the location: \( (x, y) \).
• The position vector is the arrow from the origin to that location: \(\begin{pmatrix} x \\ y \end{pmatrix}\).
• They use the same numbers, just in a different format!
2. Finding the Vector Between Two Points
One of the most useful things about position vectors is that they help us find the vector between any two points, say \( A \) and \( B \). This is called a displacement vector, written as \(\vec{AB}\).
The Logic:
To get from \( A \) to \( B \), imagine you have to go back to the origin first and then out to \( B \).
Path: \( A \rightarrow O \rightarrow B \)
In vector terms: \(\vec{AB} = \vec{AO} + \vec{OB}\)
Since \(\vec{AO}\) is just the opposite of the position vector a, we get:
The Golden Formula:
\(\vec{AB} = \mathbf{b} - \mathbf{a}\)
Step-by-Step Example:
If \( A = (1, 5) \) and \( B = (4, 2) \):
1. Write down the position vectors: \(\mathbf{a} = \begin{pmatrix} 1 \\ 5 \end{pmatrix}\) and \(\mathbf{b} = \begin{pmatrix} 4 \\ 2 \end{pmatrix}\).
2. Apply the formula \(\mathbf{b} - \mathbf{a}\):
\(\vec{AB} = \begin{pmatrix} 4 \\ 2 \end{pmatrix} - \begin{pmatrix} 1 \\ 5 \end{pmatrix} = \begin{pmatrix} 4 - 1 \\ 2 - 5 \end{pmatrix} = \begin{pmatrix} 3 \\ -3 \end{pmatrix}\).
Memory Aid: "Destination minus Departure"
To find the vector, always take the second point's vector and subtract the first point's vector. End minus Start.
Common Mistake to Avoid:
Many students accidentally do \(\mathbf{a} - \mathbf{b}\). Remember, you are going to \( B \), so \( B \) comes first in the subtraction!
Key Takeaway:
To find the vector \(\vec{AB}\), use (Position Vector of B) - (Position Vector of A).
3. Distance Between Two Points
Sometimes the question doesn't want the vector; it wants the actual distance (the length of the line) between point \( A \) and point \( B \). In vector language, this is the magnitude of the vector \(\vec{AB}\), written as \( |\vec{AB}| \).
Prerequisite Concept: Pythagoras
Calculating the distance between position vectors is just using Pythagoras' Theorem! If your vector is \(\begin{pmatrix} x \\ y \end{pmatrix}\), the distance is \(\sqrt{x^2 + y^2}\).
Process:
1. Find the displacement vector \(\vec{AB}\) first (using \(\mathbf{b} - \mathbf{a}\)).
2. Calculate the magnitude of that resulting vector.
Example:
Find the distance between \( A(1, 2) \) and \( B(4, 6) \).
• Vector \(\vec{AB} = \begin{pmatrix} 4 \\ 6 \end{pmatrix} - \begin{pmatrix} 1 \\ 2 \end{pmatrix} = \begin{pmatrix} 3 \\ 4 \end{pmatrix}\).
• Distance \( = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \) units.
Encouraging Note:
If you get a vector with negative numbers, like \(\begin{pmatrix} -3 \\ 4 \end{pmatrix}\), don't worry! When you square a negative number, it becomes positive: \( (-3)^2 = 9 \). Distance is always a positive value.
Quick Review Box:
• Vector: \(\vec{AB} = \mathbf{b} - \mathbf{a}\)
• Distance: \( |\vec{AB}| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)
4. Working with Multiple Points (Collinear Points)
Sometimes you’ll be asked to prove that three points \( A, B, \) and \( C \) lie on a straight line. This is called being collinear.
How to check:
1. Find the vector \(\vec{AB}\).
2. Find the vector \(\vec{BC}\).
3. If one vector is a scalar multiple of the other (e.g., \(\vec{AB} = 2 \times \vec{BC}\)), they are parallel. Since they both share point \( B \), they must be on the same straight line!
Analogy:
Imagine two people walking. If Person 1 walks 2 steps East and Person 2 walks 4 steps East, they are walking in the exact same direction. If they both passed through the same gate, they are on the same path.
Key Takeaway:
Points are collinear if the vectors between them are parallel (multiples of each other).
Chapter Summary
1. Position Vectors: Always start from the origin \( O \). The coordinates \( (x, y) \) become the vector \(\begin{pmatrix} x \\ y \end{pmatrix}\).
2. Finding the Path: To find the vector from \( A \) to \( B \), use \(\vec{AB} = \mathbf{b} - \mathbf{a}\).
3. Finding the Length: The distance between two points is the magnitude of the displacement vector: \(\sqrt{x^2 + y^2}\).
4. Notation: Keep your work tidy! Use bold letters for vectors and brackets for coordinates to avoid getting confused.
Great job! Vectors can feel abstract at first because we are using letters to represent movements, but once you master the "End minus Start" rule, you've conquered the hardest part of position vectors. Keep practicing!