What is a scalar product?

A new operation between vectors

So far, we have learnt how to add two vectors or multiply a vector by a number. The dot product is different: it combines two vectors u and v to give a single NUMBER (not a vector). It is denoted u.v (sometimes u . v).

Definition using angle and norms

If u and v are two non-zero vectors forming an angle theta between them, then:

u.v = ||u|| * ||v|| * cos(theta)

where ||u|| denotes the norm (length) of vector u, and cos(theta) denotes the cosine of the geometric angle formed by the two vectors (when they are placed at the same origin).

Special cases to note

Situation	Angle theta	Value of u·v
u and v in the same direction	0
u and v orthogonal	90	0
u and v in opposite directions	180	-

Practical example

Let u and v be two vectors with ||u|| = 5, ||v|| = 3, and an angle of 60 degrees between them. Then:

u·v = 5 * 3 * cos(60) = 15 * 0.5 = 7.5

The dot product of a vector with itself

When u = v, the angle theta is 0, so cos(theta) = 1. We obtain:

u·u = ||u||²

This relationship is very useful: it links the dot product to the square of a vector’s magnitude, and forms the basis for many distance calculations.

Common pitfall

Do not confuse u·v (a number) with u + v (a vector): the dot product is not represented by an arrow. Also be careful with the sign: if the angle is obtuse (between 90 and 180 degrees), cos(theta) is negative, so u·v is negative, even if the norms are positive.