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Vectors

Vectors and Transformations

Size and direction

Two things describe a vector: how big it is and which way it points. A scalar is just a number, with no direction attached
On a grid a vector is drawn as a line of the right length with an arrow showing which way it points

A single vector a drawn as an arrow along a dashed line, labelled to show that its length is its magnitude and the arrowhead is its direction, the two things that together describe it — Source: Vector diagrams by Save My Exams

The same vector can be drawn anywhere on the grid: only its length and direction matter, not where it starts

Three vectors drawn on squared paper starting at different places: a sloping up and to the right, b sloping down and to the right, and c running straight across, each a line of the right length with an arrowhead — Source: Vector diagrams by Save My Exams

Starting at A and finishing at B gives the vector written AB with an arrow above it; going back the other way gives BA, the same length reversed, so BA = −AB

Two points A and B joined by a red arrow pointing from A to B, labelled AB with an arrow above the letters, which is the notation for the vector that starts at A and finishes at B — Source: Vector diagrams by Save My Exams

Printed vectors are bold, and handwritten ones are underlined, though mark schemes do not penalise a missing underline

Column vectors

A column vector stacks the horizontal movement above the vertical one
The top number is positive for right and negative for left; the bottom number is positive for up and negative for down

A column vector with x above y, annotated to show that the top number is movement in the x direction, positive to the right and negative to the left, and the bottom number is movement in the y direction, positive up and negative down — Source: Basic vectors by Save My Exams

Add or subtract two column vectors by working on the top numbers and the bottom numbers separately

Vector addition and subtraction drawn on squared paper: a followed by b nose to tail gives the single vector a + b from the start of a to the end of b, and a followed by −b gives a − b in the same way — Source: Vector diagrams by Save My Exams

Multiplying by a scalar multiplies both numbers
- With a = (3, −2) and b = (−1, 5) written as columns, a + b = (2, 3) and a − b = (4, −7)
- 3a = (9, −6), and 2a + 3b = (6, −4) + (−3, 15) = (3, 11)
Follow the order of operations: multiply each vector by its scalar first, then add or subtract
Multiplying by a positive scalar keeps the direction and changes the length; a negative scalar reverses the direction as well

A vector a drawn on squared paper beside −a, the same length pointing the opposite way, and −2a, twice as long and also pointing the opposite way — Source: Vector diagrams by Save My Exams

Worked example

Adding and scaling column vectors

a = (3, −1) and b = (−2, 4), both written as column vectors. Work out a + b and 2a − b.

Solution:

Add or subtract the top numbers together and the bottom numbers together, separately
a + b: top 3 + (−2) = 1, bottom −1 + 4 = 3
a + b = (1, 3)
For 2a, multiply both parts of a by 2: top 6, bottom −2
Then subtract b from that: top 6 − (−2) = 8, bottom −2 − 4 = −6
2a − b = (8, −6)
Multiplying only the top number is the standard slip; a scalar multiplies every part