Calculate the scalar product using coordinates

A very handy formula

In an orthonormal coordinate system (where the two axes are perpendicular and graduated in the same unit), if u has coordinates (x; y) and v has coordinates (x'; y'), then:

u·v = x * x' + y * y'

This formula means you do not need to know the angle between the vectors: you only need to work with the coordinates.

Step-by-step example

Let u = (2, 3) and v = (-1, 4). We calculate:

u·v = 2 * (-1) + 3 * 4 = -2 + 12 = 10

Finding the norm from the coordinates

We have seen that u·u = ||u||². Using the coordinates, u·u = x² + y², so:

||u|| = √(x² + y²)

Example: for u(3; 4), ||u|| = √(3² + 4²) = √(9 + 16) = √(25) = 5.

Useful properties of the scalar product

Property	Formula
Symmetry	u·v = v·u
Bilinearity (distributivity)	u·(v + w) = u·v + u·w
Multiplication by a real number k	(k*u).v = k * (u.v)
Notable identity	(u + v).(u + v) =

This last identity is very similar to (a + b)² = a² + 2ab + b² for numbers: this is to be expected, as the scalar product behaves like multiplication in many calculations.

Common pitfall

The formula u·v = x·x’ + y·y’ only works in an ORTHONORMAL coordinate system. In any other coordinate system (with non-perpendicular axes or different units), this simple formula no longer applies: one must then revert to the definition involving angles and norms.