Why the inner product of two orthogonal vectors is zero

[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?

 

[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?

 

[flyerpower]

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

 

[Simon Bridge]

No worries - I edited to add another kind of answer while you were replying.

 

[blue_raver22]

The reason why the inner product of two orthogonal vectors is always zero is due to the definition of orthogonality itself. Orthogonality means that two vectors are perpendicular to each other, meaning that the angle between them is 90 degrees. This is true in any euclidean space, not just in R^n.

In the real vector space R^n, the inner product is defined as the product of the magnitudes of the two vectors and the cosine of the angle between them. When the angle between two vectors is 90 degrees, the cosine of that angle is zero, making the entire inner product zero.

This concept can also be extended to any euclidean space because the definition of orthogonality remains the same. In a higher dimensional space, the angle between two vectors can still be 90 degrees, making the cosine of that angle zero and resulting in a zero inner product.

In summary, the inner product of two orthogonal vectors is always zero because of the definition of orthogonality and the fact that the cosine of a 90 degree angle is always zero. This concept holds true in any euclidean space, not just in R^n.