PlanetMath: dot product

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dot product

(Definition)

Let $u=(u_1,u_2,\ldots,u_n)$ and $v=(v_1,v_2,\ldots,v_n)$ two vectors on where is a field (like $\mathbb{R}$ or $\mathbb{C}$ ). Then we define the dot product of the two vectors as:

$\displaystyle u\cdot v=u_1v_1+u_2v_2+\cdots+u_nv_n.$

Notice that $u\cdot v$ is NOT a vector but a scalar (an element from the field ).

If are vectors in $\mathbb{R}^n$ and $\vartheta$ is the angle between them, then we also have

$\displaystyle u\cdot v=\Vert u\Vert\Vert v\Vert \cos\vartheta.$

Thus, in this case, $u\perp v$ if and only if $u\cdot v=0$ .

The special case $u \cdot u$ of scalar product is the scalar square of the vector . In $\mathbb{R}^n$ it equals to the square of the length of :

$\displaystyle u \cdot u = \Vert u \Vert^2$

"dot product" is owned by drini. [ full author list (2) | owner history (1) ]

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Other names:

scalar product

Also defines:

scalar square

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Cross-references: length, square, angle, scalar, field, vectors
There are 47 references to this entry.

This is version 8 of dot product, born on 2001-10-15, modified 2006-03-10.
Object id is 239, canonical name is DotProduct.
Accessed 27958 times total.

Classification:

AMS MSC:

15A63 (Linear and multilinear algebra; matrix theory :: Quadratic and bilinear forms, inner products)

Pending Errata and Addenda

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