Paravector - Fundamental Axiom

Fundamental Axiom

For Euclidean spaces, the fundamental axiom indicates that the product of a vector with itself is the scalar value of the length squared (positive)

Writing

and introducing this into the expression of the fundamental axiom

$(\mathbf{u} + \mathbf{w})^2 = \mathbf{u} \mathbf{u} + \mathbf{u} \mathbf{w} + \mathbf{w} \mathbf{u} + \mathbf{w} \mathbf{w},$

we get the following expression after appealing to the fundamental axiom again

$\mathbf{u} \cdot \mathbf{u} + 2 \mathbf{u} \cdot \mathbf{w} + \mathbf{w} \cdot \mathbf{w} = \mathbf{u} \cdot \mathbf{u} + \mathbf{u} \mathbf{w} + \mathbf{w} \mathbf{u} + \mathbf{w} \cdot \mathbf{w},$

which allows to identify the scalar product of two vectors as

$\mathbf{u} \cdot \mathbf{w} = \frac{1}{2}\left( \mathbf{u} \mathbf{w} + \mathbf{w} \mathbf{u} \right).$

As an important consequence we conclude that two orthogonal vectors (with zero scalar product) anticommute

$\mathbf{u} \mathbf{w} + \mathbf{w} \mathbf{u} = 0$

Famous quotes containing the words fundamental and/or axiom:

“Each [side in this war] looked for an easier triumph, and a result less fundamental and astounding. Both read the same Bible, and pray to the same God; and each invokes His aid against the other. It may seem strange that any men should dare to ask a just God’s assistance in wringing their bread from the sweat of other men’s faces; but let us judge not that we be not judged.”
—Abraham Lincoln (1809–1865)

“It’s an old axiom of mine: marry your enemies and behead your friends.”
—Robert N. Lee. Rowland V. Lee. King Edward IV (Ian Hunter)