Introduction: Mapping Your World with Vectors
Welcome to the world of Position Vectors! If you have ever used a map or played a video game, you have already used the logic behind this chapter. While a general vector tells you how to move (like "walk 5 meters North"), a position vector tells you exactly where something is located relative to a fixed starting point, which we call the Origin (O).
Think of the Origin as your home. If you tell a friend your "position vector," you are telling them exactly how to get from your home to where you are standing right now. In this section, we will learn how to describe locations in 2D and 3D space and how to calculate the distances between them.
Don’t worry if this seems a bit abstract at first; once you see the patterns, it’s just like following a set of GPS coordinates!
1. What is a Position Vector?
A position vector is a vector that starts at the Origin \(O(0, 0, 0)\) and ends at a specific point, let's call it \(A\). We write this as \(\vec{OA}\) or simply as a lowercase bold letter \(\mathbf{a}\).
Key Terminology
To master this chapter, you need to be comfortable with these specific terms from the syllabus:
- Position Vector: A vector starting from the origin \(O\). If point \(A\) is at \((3, 4)\), its position vector is \(\vec{OA} = \begin{pmatrix} 3 \\ 4 \end{pmatrix}\).
- Displacement Vector: A vector that connects two points, like \(A\) and \(B\). It describes the "journey" from one point to another.
- Component Vector: The individual parts of a vector, like the \(x, y,\) and \(z\) parts. For example, in \(3\mathbf{i} + 4\mathbf{j}\), the \(3\mathbf{i}\) and \(4\mathbf{j}\) are the components.
- Resultant Vector: The vector you get when you add two or more vectors together.
- Equal Vectors: Two vectors are equal if they have the same magnitude (length) and direction.
- Parallel Vectors: Vectors that point in the same (or exactly opposite) direction. One is always a "scalar multiple" of the other (e.g., \(\mathbf{a}\) and \(2\mathbf{a}\) are parallel).
- Unit Vector: A vector with a magnitude of exactly 1. We use \(\mathbf{i}, \mathbf{j},\) and \(\mathbf{k}\) as standard unit vectors for the \(x, y,\) and \(z\) axes.
Quick Review: Every point in space has a unique position vector. If point \(P\) is \((x, y, z)\), then \(\vec{OP} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k}\).
2. Finding the Vector Between Two Points
This is the most "magical" formula in vectors! If you know where point \(A\) is (its position vector \(\mathbf{a}\)) and where point \(B\) is (its position vector \(\mathbf{b}\)), how do you find the vector that goes from \(A\) to \(B\)?
The Displacement Vector \(\vec{AB}\) is found using:
\[\vec{AB} = \vec{OB} - \vec{OA} = \mathbf{b} - \mathbf{a}\]
Memory Aid: "Destination minus Start"
To find the vector between two points, always subtract the start from the finish.
Going from \(A\) to \(B\)? It’s \(B - A\).
Going from \(P\) to \(Q\)? It’s \(Q - P\).
Example:
Point \(A\) has position vector \(\mathbf{a} = 2\mathbf{i} + 3\mathbf{j}\) and point \(B\) has position vector \(\mathbf{b} = 5\mathbf{i} - \mathbf{j}\).
The vector \(\vec{AB} = \mathbf{b} - \mathbf{a}\)
\(\vec{AB} = (5\mathbf{i} - \mathbf{j}) - (2\mathbf{i} + 3\mathbf{j})\)
\(\vec{AB} = 3\mathbf{i} - 4\mathbf{j}\)
Key Takeaway: Subtracting position vectors gives you the displacement vector (the "directions") to get from one point to another.
3. Calculating Distance Between Points
Once you have the displacement vector \(\vec{AB}\), you can find the actual distance (the length of the line) between point \(A\) and point \(B\). This is simply the magnitude of the vector.
The Formula
If \(\vec{OA} = \begin{pmatrix} a \\ b \end{pmatrix}\) and \(\vec{OB} = \begin{pmatrix} c \\ d \end{pmatrix}\), the distance is:
\[\text{Distance} = |\vec{AB}| = \sqrt{(c-a)^2 + (d-b)^2}\]
In 3D, we just add the third component:
\[\text{Distance} = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}\]
Did you know? This is actually just Pythagoras' Theorem! We are finding the hypotenuse of a right-angled triangle where the sides are the differences in coordinates.
Common Mistake to Avoid:
When squaring negative numbers (like \((-3)^2\)), remember the result is always positive (\(9\)). Students often accidentally subtract these squares on their calculators!
4. Summary Table for Quick Revision
Term / Concept | Mathematical Notation
-----------------|--------------------------
Position Vector of \(A\) | \(\vec{OA}\) or \(\mathbf{a}\)
Vector from \(A\) to \(B\) | \(\mathbf{b} - \mathbf{a}\)
Distance between \(A\) and \(B\) | \(|\mathbf{b} - \mathbf{a}|\)
Parallel to \(\mathbf{a}\) | \(\lambda \mathbf{a}\) (where \(\lambda\) is a constant)
Unit vector in direction \(\mathbf{a}\) | \(\frac{\mathbf{a}}{|\mathbf{a}|}\)
5. Final Tips for Success
- Draw a diagram: Even a messy sketch of the Origin and points \(A\) and \(B\) can help you visualize the "Destination minus Start" rule.
- Check your notation: When writing by hand, always underline your vectors (e.g., \(\underline{a}\)) to show they aren't just regular numbers.
- 3D is the same as 2D: Don't be intimidated by the \(\mathbf{k}\) component. Every rule you learned for 2D vectors works exactly the same way in 3D!
Key Takeaway: Position vectors are the anchor points of coordinate geometry. Master the \(\mathbf{b} - \mathbf{a}\) subtraction and the magnitude formula, and you will have the foundation needed for the rest of the Vectors section!