Dot Product - Properties

Properties

The following properties hold if a, b, and c are real vectors and r is a scalar.

The dot product is commutative:

The dot product is distributive over vector addition:

The dot product is bilinear:

When multiplied by a scalar value, dot product satisfies:

(these last two properties follow from the first two).

Two non-zero vectors a and b are orthogonal if and only if a⋅b = 0.

Unlike multiplication of ordinary numbers, where if ab = ac, then b always equals c unless a is zero, the dot product does not obey the cancellation law:

If a⋅b = a⋅c and a ≠ 0, then we can write: a ⋅ (b − c) = 0 by the distributive law; the result above says this just means that a is perpendicular to (b − c), which still allows (b − c) ≠ 0, and therefore b ≠ c.

Provided that the basis is orthonormal, the dot product is invariant under isometric changes of the basis: rotations, reflections, and combinations, keeping the origin fixed. The above mentioned geometric interpretation relies on this property. In other words, for an orthonormal space with any number of dimensions, the dot product is invariant under a coordinate transformation based on an orthogonal matrix. This corresponds to the following two conditions:

• The new basis is again orthonormal (i.e., it is orthonormal expressed in the old one).
• The new base vectors have the same length as the old ones (i.e., unit length in terms of the old basis).

If a and b are functions, then the derivative of a ⋅ b is a'⋅b + a⋅b'