# dot product

Let $u=({u}_{1},{u}_{2},\mathrm{\dots},{u}_{n})$ and $v=({v}_{1},{v}_{2},\mathrm{\dots},{v}_{n})$ two vectors on ${k}^{n}$ where $k$ is a field (like $\mathbb{R}$ or $\u2102$).
Then we define the *dot product ^{}* of the two vectors as:

$$u\cdot v={u}_{1}{v}_{1}+{u}_{2}{v}_{2}+\mathrm{\cdots}+{u}_{n}{v}_{n}.$$ |

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

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

$$u\cdot v=\parallel u\parallel \parallel v\parallel \mathrm{cos}\vartheta .$$ |

Thus, in this case, $u\u27c2v$ if and only if $u\cdot v=0$.

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

$$u\cdot u={\parallel u\parallel}^{2}$$ |

Title | dot product |

Canonical name | DotProduct |

Date of creation | 2013-03-22 11:46:33 |

Last modified on | 2013-03-22 11:46:33 |

Owner | drini (3) |

Last modified by | drini (3) |

Numerical id | 13 |

Author | drini (3) |

Entry type | Definition |

Classification | msc 83C05 |

Classification | msc 15A63 |

Classification | msc 14-02 |

Classification | msc 14-01 |

Synonym | scalar product |

Related topic | CauchySchwarzInequality |

Related topic | CrossProduct |

Related topic | Vector |

Related topic | DyadProduct |

Related topic | InvariantScalarProduct |

Related topic | AngleBetweenLineAndPlane |

Related topic | TripleScalarProduct |

Related topic | ProvingThalesTheoremWithVectors |

Defines | scalar square |