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:
(3)
\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 $\vec v$:
innerproduct.jpg
Bibliography
1. J.E.Marsden, A.Tromba, "Vector Calculus",Freeman, 5th ed., 2003. http://www.amazon.com/Vector-Calculus-Jerrold-Marsden/dp/0716749920
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