Why the inner product of two orthogonal vectors is zero

1. Dec 11, 2011

flyerpower

Why is the inner product of two orthogonal vectors always zero?

For example, in the real vector space R^n, the inner product is defined as ||a|| * ||b|| * cos(theta), and if they are orthogonal, cos(theta) is zero.
I can understand that, but how does this extend to any euclidean space?

Last edited: Dec 11, 2011
2. Dec 11, 2011

Simon Bridge

The inner product is just the amount that one vector extends in the direction of the other - it's how far one arrow "leans over" the other. If they are orthogonal, then they don't lean over each other at all. That's what "orthogonal means - they have zero extent in each other's directions.

This generalizes to many dimensions because each vector defines a direction in that space and the other one may have some lean in that direction.

Algebraically, if u and v are vectors, then their scalar product is defined as $u^t \cdot v$ - in terms of components in an n-D basis that would be:

$$\left ( u_1,u_2, \ldots ,u_n \right ) \left ( \begin{array}{c} v_1\\ v_2 \\ \vdots \\ v_n \end{array}\right )$$

Try it for any two orthogonal vectors ... it's obvious for any two basis vectors...

$$\left ( 1,0, \ldots , 0 \right ) \left ( \begin{array}{c} 0\\ 0 \\ \vdots \\ 1 \end{array}\right )$$

see?

Last edited: Dec 11, 2011
3. Dec 11, 2011

flyerpower

Thank you, that's the kind of answer i was looking for :)

4. Dec 11, 2011