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'dot product'
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| Title of object: |
dot product |
| Canonical Name: |
DotProduct |
| Type: |
Definition |
| Created on: |
2001-10-15 23:22:08 |
| Modified on: |
2003-10-19 23:03:20 |
| Classification: |
msc:15A63 |
| Synonyms: |
dot product=scalar product |
Revision comment (for changes between this and next version):
| Changes for correction #4130 ('spelling'). |
Preamble:
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{graphicx}
\usepackage{xypic} |
Content:
Let $u=(u_1,u_2,\ldots,u_n)$ and $v=(v_1,v_2,\ldots,v_n)$ two vectors on $k^n$ where $k$ is a field (like $\mathbb{R}$ or $\mathbb{C}$).
Then we define the \emph{dot product} of the two vectors as:
$$u\cdot v=u_1v_1+u_2v_2+\cdots+u_nv_n.$$
Notice that $u\cdot v$ is NOT a vector but an scalar (an element from the field $k$).
If $u,v$ are vectors in $\mathbb{R}^n$ and $\theta$ is the angle between them, then we also have
$$u\cdot v=\Vert u\Vert\Vert v\Vert \cos\theta.$$
Thus, in this case, $u\perp v$ if and only if $u\cdot v=0$. |
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