Scalar product: definitions and properties

Analytical definition

In an orthonormal coordinate system in space, the scalar product of u(x₁, y₁, z₁) and v(x₂, y₂, z₂) is the real number: u · v = x₁ · x₂ + y₁ · y₂ + z₁ · z₂

Geometric definition

We also have u·v = ||u|| · ||v|| · cos(θ), where θ is the angle formed by u and v. This dual definition allows us to calculate an angle from the coordinates: cos(θ) = (u·v) / (||u|| · ||v||)

Example: u(1, 0, 1), v(0, 1, 1). u·v = 0 + 0 + 1 = 1. ||u|| = √2, ||v|| = √2. cos(theta) = 1/2, so theta = π/3 (60 degrees).

Properties

• Symmetry: u·v = v·u
• Bilinearity: u·(v+w) = u·v + u·w and (k·u)·v = k·(u·v)
• u·u = ||u||² ≥ 0
• u·v = 0 if and only if u and v are orthogonal (or one of them is zero)

Property	Statement
Symmetry	u·v = v·u
Bilinearity	u·(v + w) = u·v + u·w
Norm	u·u =
Orthogonality	u·v = 0 ⇔ u is perpendicular to v

Common pitfall

The dot product yields a number (a scalar), never a vector: writing “u·v = (2,3,1)” is a common mistake at the start of the academic year. Do not confuse this with the cross product, which returns a vector.