Coordinates of a vector

Coordinates of a vector

In a coordinate system (O; →i, →j), every vector →u can be uniquely expressed as →u = x→i + y→j. The pair (x; y) is called the coordinates of ->u, denoted ->u(x; y).

If A(xA; yA) and B(xB; yB) are two points, then the vector ->AB has the coordinates:

->AB (xB – xA; yB – yA)

Norm of a vector

The norm of the vector ->u(x; y) is calculated using the Pythagorean theorem:

||u|| = sqrt(x² + y²)

Example: if ->u(3; 4), then ||u|| = √(9 + 16) = √(25) = 5.

Coordinates of the midpoint

The midpoint I of the line segment [AB] has the following coordinates:

I( (xA + xB)/2 ; (yA + yB)/2 )

Operations on coordinates

Operation	Formula
Sum of ->u and ->v	(x + x'; y + y')
Product by k	(kx; ky)
Equality	->u = ->v <=> x = x' and y = y'

Common pitfall

To calculate the coordinates of ->AB, we always use ‘destination coordinates minus starting coordinates’: (xB – xA; yB – yA), and never the other way round.