# difference of vectors

Let $\overrightarrow{a}$ and $\overrightarrow{b}$ be two vectors in the plane (or in a vector space^{}). The difference vector or difference $\overrightarrow{a}-\overrightarrow{b}$ of $\overrightarrow{a}$ and $\overrightarrow{b}$ is a vector $\overrightarrow{d}$ such that

$$\overrightarrow{b}+\overrightarrow{d}=\overrightarrow{a}.$$ |

Thus we have

$\overrightarrow{b}+(\overrightarrow{a}-\overrightarrow{b})=\overrightarrow{a}.$ | (1) |

According to the procedure of forming the sum of vectors by setting the addends one after the other, the equation (1) tallies with the picture below; when the minuend and the subtrahend emanate from a common initial point, their difference vector can be directed from the terminal point of the subtrahend to the terminal point of the minuend.

Remark. It is easily seen that the difference $\overrightarrow{a}-\overrightarrow{b}$ is same as the sum vector

$$\overrightarrow{a}+(-\overrightarrow{b})$$ |

where $-\overrightarrow{b}$ is the opposite vector of $\overrightarrow{b}$: it may be represented by the directed line segment from the terminal point of $\overrightarrow{b}$ to the initial point of $\overrightarrow{b}$.

Title | difference of vectors |

Canonical name | DifferenceOfVectors |

Date of creation | 2013-03-22 17:47:19 |

Last modified on | 2013-03-22 17:47:19 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 9 |

Author | pahio (2872) |

Entry type | Definition |

Classification | msc 53A45 |

Synonym | vector difference |

Synonym | vector subtraction |

Related topic | CommonPointOfTriangleMedians |

Related topic | Difference2 |

Related topic | ProvingThalesTheoremWithVectors |

Related topic | VectorValuedFunction2 |

Defines | difference vector |

Defines | opposite vector |