The Big Idea
A vector is a quantity that describes both how far (magnitude) and which way (direction) something moves or exists. Unlike a plain number (scalar), vectors capture movement in 2D space and can be added, multiplied, and used to solve real geometric problems.
Quick Summary
Column Vectors
Two numbers (x, y) showing horizontal and vertical movement from a point.
Adding & Subtracting
Combine components separately: add/subtract tops, add/subtract bottoms.
Magnitude (Length)
Use Pythagoras: |a| = √(x² + y²) — always positive.
Position Vectors
Describe location relative to origin O. Components = coordinates.
Displacement Vectors
Movement between two points: AB = B's position − A's position.
Parallel Vectors
One is a scalar multiple of the other: b = ka (same direction if k > 0).
Introduction to Column Vectors
What is a Column Vector?
A column vector is a way of writing instructions for moving from one point to another. It's just two numbers stacked vertically.
Example
The column vector (6) means: 6 units to the right and 3 units up.
(3)
• The top number tells you the horizontal movement (x-direction): positive = right, negative = left.
• The bottom number tells you the vertical movement (y-direction): positive = up, negative = down.
Vector Notation
When typed, vectors are written in bold: a
When handwritten, they're underlined: a̲
This shows they're vectors, not just plain numbers (scalars).
Adding Column Vectors
To add two column vectors, add the top components together and the bottom components together separately.
Vector Addition
(a) + (c) = (a + c)
(b) (d) (b + d)
Worked Example
(5) + (3) = (8)
(2) (−1) (1)
Why? Top: 5 + 3 = 8. Bottom: 2 + (−1) = 1.
Subtracting Column Vectors
Subtract the second vector's components from the first vector's components.
Vector Subtraction
(a) − (c) = (a − c)
(b) (d) (b − d)
Worked Example
(5) − (3) = (2)
(2) (−1) (3)
Why? Top: 5 − 3 = 2. Bottom: 2 − (−1) = 2 + 1 = 3.
Multiplying a Vector by a Scalar (Number)
When you multiply a vector by a number (called a scalar), you multiply both components by that number. This changes the vector's length but keeps it pointing in the same direction (unless the scalar is negative).
Scalar Multiplication
k × (x) = (k·x)
(y) (k·y)
Worked Example
3 × (2)
(6)
(−1)
(−3)
Multiply both parts by 3:
Scalar ≠ Vector
Writing Expressions as a Single Column Vector
2(5/2) + 5(3/−1)
Worked Example
Simplify: