Introduction to Plane Vector Geometry

Welcome! In this chapter, we are going to explore vectors. While regular numbers (which mathematicians call scalars) tell us "how much" of something there is, vectors are much more exciting because they tell us two things at once: how far and in which direction.

Think of it like giving someone directions. If you say "walk 50 metres," they won't know where to go. But if you say "walk 50 metres North," you’ve given them a vector! Vectors are essential for everything from GPS navigation in your phone to how video game characters move across a screen.

Don't worry if this seems a bit abstract at first. We will break it down into simple steps, using grids and easy calculations to make you a vector expert.


What is a Vector?

A vector is a quantity that has both magnitude (size) and direction. In geometry, we usually represent vectors as arrows. The length of the arrow shows the magnitude, and the way the arrow points shows the direction.

Key Terms:

  • Scalar: A simple number with size only (like 5kg or 10 minutes).
  • Vector: A movement with size and direction (like 3 steps right and 2 steps up).
  • Magnitude: The "length" or "size" of the vector.

Quick Review: We usually name a vector using bold letters like a or by the start and end points with an arrow on top, like \(\vec{AB}\) (which means the movement from point A to point B).


Column Vectors

The easiest way to write a vector in 2D is as a column vector. It looks like two numbers inside a tall bracket:

\(\begin{pmatrix} x \\ y \end{pmatrix}\)

  • The top number (x) tells you how far to move horizontally. Positive is Right, negative is Left.
  • The bottom number (y) tells you how far to move vertically. Positive is Up, negative is Down.

Example: The vector \(\begin{pmatrix} 3 \\ -2 \end{pmatrix}\) means "move 3 squares to the right and 2 squares down."

Drawing Vectors on a Grid

When drawing a vector, you can start anywhere! A vector isn't a fixed point; it's a movement. Just pick a starting spot, count the squares, and draw your arrow.

Did you know? Two vectors are considered equal if they have the same size and point in the same direction, even if they start in different places on the grid!

Common Mistake to Avoid: Don't confuse column vectors with coordinates. Coordinates \((x, y)\) are a location; a column vector \(\begin{pmatrix} x \\ y \end{pmatrix}\) is a journey.


Vector Arithmetic

We can add, subtract, and multiply vectors just like regular numbers, but we do it piece by piece.

1. Scalar Multiplication

This is when you multiply a vector by a regular number (a scalar). It’s like "zooming in" or "zooming out" on the movement. You simply multiply both the top and bottom numbers by that scalar.

Example: If a = \(\begin{pmatrix} 2 \\ 5 \end{pmatrix}\), then 3a = \(\begin{pmatrix} 3 \times 2 \\ 3 \times 5 \end{pmatrix} = \begin{pmatrix} 6 \\ 15 \end{pmatrix}\).

Memory Aid: If you multiply a vector by a negative number, the arrow flips and points in the opposite direction!

2. Adding Vectors

Adding vectors is like taking a "multi-stage" journey. If you follow vector a and then follow vector b, the total journey is a + b. To calculate it, just add the top numbers together and the bottom numbers together.

\(\begin{pmatrix} 2 \\ 3 \end{pmatrix} + \begin{pmatrix} 4 \\ 1 \end{pmatrix} = \begin{pmatrix} 2+4 \\ 3+1 \end{pmatrix} = \begin{pmatrix} 6 \\ 4 \end{pmatrix}\)

The "Nose-to-Tail" Rule: When drawing, place the start (tail) of the second vector at the end (nose) of the first vector. The "resultant" vector goes from the very start to the very end.

3. Subtracting Vectors

Subtracting b is the same as adding negative b. Mathematically, just subtract the top numbers and subtract the bottom numbers.

\(\begin{pmatrix} 5 \\ 7 \end{pmatrix} - \begin{pmatrix} 2 \\ 3 \end{pmatrix} = \begin{pmatrix} 5-2 \\ 7-3 \end{pmatrix} = \begin{pmatrix} 3 \\ 4 \end{pmatrix}\)


Using Vectors in Geometry (Higher Tier)

Sometimes, we use vectors to prove things about shapes like parallelograms or triangles. This is where we look at "routes" around a shape.

Parallel Vectors

Two vectors are parallel if one is a multiple of the other. For example, \(\begin{pmatrix} 1 \\ 2 \end{pmatrix}\) and \(\begin{pmatrix} 3 \\ 6 \end{pmatrix}\) are parallel because the second one is just the first one multiplied by 3.

Key Takeaway: If vector p = \(k\)q (where \(k\) is any number), then p and q are parallel.

Finding Routes

If you are given a shape where sides are labeled as vectors (like a and b), you can find the vector for any other path by following the sides.

  • If you go with the arrow, the vector is positive.
  • If you go against the arrow, the vector is negative.

Example: In a triangle ABC, if \(\vec{AB} = \mathbf{a}\) and \(\vec{BC} = \mathbf{b}\), then the shortcut from A to C (\(\vec{AC}\)) is simply a + b.


Summary Checklist

Before you move on, make sure you can:

  • Write a movement as a column vector \(\begin{pmatrix} x \\ y \end{pmatrix}\).
  • Draw a vector correctly on a grid using "Right/Left" and "Up/Down."
  • Multiply a vector by a number (scalar multiplication).
  • Add and Subtract column vectors by calculating the top and bottom rows separately.
  • Identify parallel vectors (one is a multiple of the other).

Final Encouragement: Vectors are just a way of describing a journey. If you can count squares on a grid and do basic addition, you can do vectors! Keep practicing drawing them, and the rules will soon become second nature.