Level 5 · Ages 15–16
Read it again any time, including while you work through the exercises.
Remember translations, those slides that move a whole figure in one motion? The vector is precisely the mathematical object encoding that motion. Picture an arrow: it has a direction (the line it follows), an orientation (which way it points) and a norm (its length). That is a vector!
Key point: a vector has no position. The arrow from A to B and the one from C to D represent the SAME vector if they share direction, orientation and length. A vector is a pure move: 3 right and 2 up, no matter where you start.
In a coordinate system, everything becomes computable. The vector AB going from A(xA, yA) to B(xB, yB) has coordinates (xB - xA, yB - yA): arrival minus departure, always in that order! Its coordinates say exactly how far you move horizontally and vertically.
Two formulas to know by heart. The midpoint of segment [AB] has coordinates equal to the averages: ((xA + xB)/2, (yA + yB)/2). And the norm of a vector with coordinates (x, y) comes from Pythagoras: it is the square root of x² + y², because the horizontal and vertical moves form a right triangle.
To add two vectors, chain the moves: simply add the coordinates. Vectors are the perfect tool to describe journeys, forces or speeds: they will follow you for a long time!
In a coordinate system, A(1, 2) and B(4, 7). Find the coordinates of vector AB.
x = xB - xA = 4 - 1 = 3
Arrival minus departure for the x-coordinates: from 1 to 4, I move 3 to the right.
y = yB - yA = 7 - 2 = 5
Same for the y-coordinates: from 2 to 7, I go up 5. Vector AB encodes the move (3, 5).
Result: AB has coordinates (3, 5).
A(2, 4) and B(6, 10). Find the coordinates of M, the midpoint of [AB].
xM = (2 + 6) / 2 = 4
The midpoint is halfway: I take the average of the x-coordinates, (2 + 6) divided by 2.
yM = (4 + 10) / 2 = 7
Then the average of the y-coordinates: (4 + 10) divided by 2 equals 7. The midpoint is at (4, 7).
Result: M(4, 7).