Euclidean Inner Product

## Definition

For a regular vectors $\vec a$ and $\vec b$ in Euclidean space the inner product is defined by:

(1)\begin{align} \vec a\cdot\vec b = {a_1}{b_1} + {a_2}{b_2} + ... + {a_n}{b_n} \end{align}

## Additional info

The (smaller) angle θ between two vectors $\vec a$ and $\vec b$ is computed by:

(2)\begin{align} cos\left( \theta \right) = \frac{{\vec a\cdot\vec b}}{{‖\vec a‖‖\vec b‖}} \end{align}

where ‖.‖ is an Euclidean Norm.

*Geometric interpretations [1]*

- The vector of the orthogonal projection in the direction $\vec v$ is defined by:

\begin{align} proj(\vec a) = \frac{{\vec a\cdot\vec v}}{{‖\vec v‖^2}}\vec v \end{align}

- If $\vec v$ is the unit vector (‖$\vec v$‖=1) , then $\vec a$·$\vec v$ is the length of projection of vector $\vec a$ to the direction $\vec v$:

Bibliography

1. J.E.Marsden, A.Tromba, "Vector Calculus",Freeman, 5th ed., 2003. http://www.amazon.com/Vector-Calculus-Jerrold-Marsden/dp/0716749920

page revision: 5, last edited: 19 Jun 2011 10:52