# 10.3 - The Dot Product

- StudyBlue
- North-carolina
- University of North Carolina - Charlotte
- Mathematics
- Mathematics 2241
- Gordon
- 10.3 - The Dot Product

**Created:**2016-12-20

**Last Modified:**2017-01-18

**a**=〈a

_{1}, a

_{2}, a

_{3}〉and

**b**=〈b

_{1}, b

_{2}, b

_{3}〉, then the

**dot product**of

**a**&

**b**is the number

**a•b**given by

**a•b:**

- a
_{1}b_{1}+ a_{2}b_{2}+ a_{3}b_{3} - a
_{1}b_{1}− a_{2}b_{2}− a_{3}b_{3}

_{1}b

_{1}

**+**a

_{2}b

_{2}

**+**a

_{3}b

_{3}

- vector
- scalar

_{1}, a

_{2}〉•〈b

_{1}, b

_{2}〉=

- a
_{1}b_{1}+ a_{2}b_{2} - a
_{1}b_{1}− a_{2}b_{2}

_{1}b

_{1}+ a

_{2}b

_{2}

**Properties Of the Dot Product**: If

**a**,

**b**, &

**c**are vectors in V

_{3}&

*c*is scalar, then:

**a • a**=

- 2
**a** - |
**a**|^{2}

**a**|

^{2}

**Properties Of the Dot Product**: If

**a**,

**b**, &

**c**are vectors in

*V*&

_{3}*c*is scalar, then:

**a**•

**b**=

**b**•**a**- |
**a**|^{2}

**b**•

**a**

**Properties Of the Dot Product**: If

**a**,

**b**, &

**c**are vectors in

*V*&

_{3}*c*is scalar, then:

**a •**(

**b**+

**c**) =

- (
**a**+**b**)**c**

**a**•**b**+**a**•**c**

**a**•

**b**+

**a**•

**c**

**Properties Of the Dot Product**: If

**a**,

**b**, &

**c**are vectors in V

_{3}&

*c*is scalar, then: (

*c*

**a**) •

**b**=

*c*(**a • b**) =**a •**(*c***b**)*c***ab**

*c*(

**a**•

**b**) =

**a**• (

*c*

**b**)

**Properties Of the Dot Product**: If

**a**,

**b**, &

**c**are vectors in V

_{3}&

*c*is scalar, then: 0

**• a**=

**a****0**

- c
^{2}= a^{2}+ b^{2} - a
^{2}= b^{2}+ c^{2}sinθ - a
^{2}= b^{2}+ c^{2}− 2bccosθ

^{2}= b

^{2}+ c

^{2}− 2bccos

*θ*

*θ*is pointing at.

*θ*is the angle between the vectors

**a**and

**b**, then:

**a • b**= |**a**| |**b**| cos*θ***a • b**= |**a**| |**b**| sin*θ*

**a • b**= |

**a**| |

**b**| cos

*θ*

*θ*is the angle between the nonzero vectors

**a**and

**b**, then:

- cos
*θ*= (**a • b**) / (**a • b**) - cos
*θ*= (**a****•****b**) / (|**a**|**•**|**b**|)

*θ*= (

**a • b**) / (|

**a**|

**•**|

**b**|)

**a**and

**b**are _________ if and only if

**a • b**=

**0**

- parallel
- orthogonal

**a • b**is ________ if

**a**and

**b**point in the same general direction

- positive
- negative
- zero

**a • b**is ________ if

**a**and

**b**are perpendicular (orthogonal).

- positive
- negative
- zero

**a • b**is ________ if

**a**and

**b**point in general opposite directions.

- positive
- negative
- zero

**a**and

**b**point in exactly the same direction, we have

*θ*= 0, so cos

*θ*= 1 and

**a**•**b**= |**a**| |**b**| sin*θ***a**•**b**= |**a**| |**b**|

**a**•

**b**= |

**a**| |

**b**|

**a**and

**b**point in exactly opposite directions, then

*θ*= π and so cos

*θ*= −1 and

**a**•

**b**= |

**a**| |

**b**|

**a**•

**b**= −|

**a**| |

**b**|

**a**•

**b**= −|

**a**| |

**b**|

**a**and

**b**are called ___________ or

**orthogonal**if the angle between them is

*θ*= π/2.

- perpendicular
- parallel

**a**•**b**= |**a**| |**b**| cos (π/2) = 0**a**•**b**= |**a**| |**b**| cos (0) =**a**•**b**= |**a**| |**b**|

**a**•

**b**= |

**a**| |

**b**| cos (π/2) = 0

**a**•

**b**> 0

*θ*= acute

*θ*= obtuse

*θ*= π/2

**a**•

**b**= 0

*θ*= acute

*θ*= obtuse

*θ*= π/2

**a**•

**b**< 0

*θ*= acute

*θ*= obtuse

*θ*= π/2

_{a}

**b**

**b**).

- comp
_{a}**b**= (**a • b**) / |**a|** - proj
_{a}**b**= ((**a • b**) / |**a**|) (**a**/ |**a**| ) = ( (**a • b**) / |**a**|^{2})**a**

_{a}

**b**= (

**a • b**) / |

**a**|

- comp
_{a}**b**= (**a • b**) / |**a**| - proj
_{a}**b**= ((**a • b**) / |**a**|) (**a**/ |**a**| ) = ( (**a • b**) / |**a**|^{2})**a**

_{a}

**b**= ((

**a • b**) / |

**a**|) (

**a**/ |

**a**| ) = ( (

**a • b**) / |

**a**|

^{2})

**a**

**a**.

- vector
- scalar

**Scalar proj.**= comp

_{a}

**b**= (

**a • b**) / |

**a**|

**Unit vector**=

**a**/ |

**a**|

**a • b**) / |

**a**|) (

**a**/ |

**a**| ) = ( (

**a • b**) / |

**a**|

^{2})

**a**

**W**= ( |

**F**| cos

*θ*) |

**D**|

**W**= |**F**| |**D**| sin*θ*=**F • D****W**= |**F**| |**D**| cos*θ*=**F • D**

**W**= |

**F**| |

**D**| cos

*θ*=

**F • D**

**F**is the dot product

**F•D**, where

**D**is the displacement vector.

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