What is a vector? Column vectors and translations
A scalar has only a size, like 5 kg or 20 °C. A vector has a size and a direction, like "3 m to the right and 2 m up".
On a square grid we write a vector as a column vector: the top number is the move across (right is +, left is −) and the bottom number is the move up (up is +, down is −). So (3 / 2) means 3 right, 2 up, and (−2 / −1) means 2 left, 1 down. (On paper the two numbers are stacked in tall brackets.)
We name vectors with a bold or underlined small letter, a, or with two capital letters and an arrow, AB→, meaning "from A to B".
A translation slides a shape without turning it. It is described by one column vector: every corner moves by the same amount.
Adding, subtracting and scaling vectors
Adding
Put the vectors tip to tail: do a, then b from where a ended. The resultant a + b goes straight from the start to the end. In numbers, add the tops and add the bottoms: (3 / 1) + (1 / 3) = (4 / 4). Order does not matter: a + b = b + a.
Multiplying by a number (scalar)
ka multiplies both parts by k. If k = 2 the arrow is twice as long in the same direction. If k = −1 it has the same length but points the opposite way. Vectors that are multiples of each other are parallel.
Subtracting
a − b = a + (−b): reverse b, then add. In numbers, subtract the tops and the bottoms: (5 / 2) − (1 / 4) = (4 / −2).
Length (magnitude)
The length of (x / y) is |a| = √(x² + y²), by Pythagoras. For (3 / 4) the length is 5.
Vector geometry proofs
In a proof we are given a few vectors, for example OA = a and OB = b, and we write other lines using them.
- Find a path: to go from A to B, go A → O → B. So AB = AO + OB = −a + b = b − a.
- Fractions of a line: if M is the midpoint of AB, AM = ½AB. So OM = OA + AM = a + ½(b − a) = ½(a + b).
- Parallel: if PQ = 2(b − a) then PQ = 2AB, so PQ is parallel to AB and twice as long.
- Points on a straight line (collinear): if PQ = k × QR and they share point Q, then P, Q and R lie on one straight line.
Always simplify fully and say in words what the result shows.
Key formulas and definitions
- Column vector (x / y): x = across (right +), y = up (+)
- (a / b) + (c / d) = (a + c / b + d)
- k(x / y) = (kx / ky)
- a − b = a + (−b)
- |(x / y)| = √(x² + y²)
- AB = OB − OA = b − a; midpoint of AB: OM = ½(a + b)
- Parallel: one vector is a number times the other
Worked examples
1. Write the move "4 left and 1 up" as a column vector.
Left is negative across, up is positive: (−4 / 1).
2. a = (2 / 5), b = (3 / −1). Find a + b.
(2 + 3 / 5 + (−1)) = (5 / 4).
3. a = (2 / 5), b = (3 / −1). Find 3a − 2b.
3a = (6 / 15); 2b = (6 / −2). 3a − 2b = (6 − 6 / 15 − (−2)) = (0 / 17).
4. Find the length of (6 / −8).
√(6² + (−8)²) = √(36 + 64) = √100 = 10.
5. Triangle P is translated by (−3 / 2). Corner (5, 1) moves where?
x: 5 − 3 = 2; y: 1 + 2 = 3. New corner (2, 3).
6. OA = a, OB = b. M is the midpoint of AB. Show OM = ½(a + b).
AB = −a + b = b − a. AM = ½(b − a). OM = OA + AM = a + ½b − ½a = ½a + ½b = ½(a + b).
7. OP = 2a, OQ = 2b, X is the midpoint of OP and Y the midpoint of OQ. Prove XY is parallel to PQ.
PQ = −2a + 2b = 2(b − a). OX = a, OY = b, so XY = −a + b = b − a. PQ = 2 XY, so PQ is parallel to XY and twice as long.
Common mistakes
- Swapping the numbers: the top number is across, the bottom number is up.
- Forgetting the sign: going left or down gives a negative number.
- Writing AB = a − b when OA = a and OB = b. It is AB = b − a (end minus start).
- Adding lengths instead of vectors: |a + b| is usually not |a| + |b|.