# 10.4 - The Cross Product

- StudyBlue
- North-carolina
- University of North Carolina - Charlotte
- Mathematics
- Mathematics 2241
- Gordon
- 10.4 - The Cross Product

**Created:**2016-12-21

**Last Modified:**2016-12-23

**cross product**

**a**×

**b**of two vectors

**a**and

**b**, unlike the dot product, is a ________. For this reason it is also called the ________ product.

- scalar
- vector

**a**=〈a

_{1}, a

_{2}, a

_{3}〉and

**b**=〈b

_{1}, b

_{2}, b

_{3}〉, then the

**cross product**of

**a**and

**b**is the vector

**• a**×

**b**= 〈

*a*

_{2}

*b*

_{3}−

*a*

_{3}

*b*

_{2 }

**,**

*a*

_{3}

*b*

_{1}−

*a*

_{1}

*b*

_{3 }

**,**

*a*

_{1}

*b*

_{2}−

*a*

_{2}

*b*

_{1}〉

**• a**×

**b**= 〈

*a*

_{2}

*b*

_{3}+ a

_{3}

*b*

_{2},

*a*

_{3}

*b*

_{1}+

*a*

_{1}

*b*

_{3},

*a*

_{1}

*b*

_{2}+

*a*

_{2}

*b*

_{1}〉

**a × b**= 〈

*a*〉

_{2}b_{3 }− a_{3}b_{2}, a_{3}b_{1}− a_{1}b_{3}, a_{1}b_{2}− a_{2}b_{1}**a**=

*a*

_{1}

**i**+

*a*

_{2}

**j**+

*a*

_{3}

**k**

**b**=

*b*

_{1}

**i**+

*b*

_{2}

**j**+

*b*

_{3}

**k**

**a**×

**b**)

**cross product**is given by the following theorem.

**a**×

**b**is __________ to both

**a**and

**b**.

- orthogonal
- parallel

**a**×

**b**) •

**a**= 0

**a**×

**b**) •

**b**= 0

**a**×

**b**is orthogonal to both __ and __

**a**and**a****a**and**b**

**a**and

**b**

**a**×

**b**|

*θ*is the angle between

**a**and

**b**(so 0 ≤

*θ*≤ π), then

- |
**a**×**b**| = |**a**| |**b**| sin*θ* - |
**a**×**b**| = |**a**| |**b**| cos*θ*

**a**×

**b**| = |

**a**| |

**b**| sin

*θ*

**a**•

**b**|

*θ*is the angle between

**a**and

**b**(so 0 ≤

*θ*≤ π), then

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

**a**•

**b**| = |

**a**| |

**b**| cos

*θ*

**a**and

**b**are parallel if and only if

**a**×**b**= 1**a**×**b**= 0

**a**×

**b**= 0

**cross product**of two nonzero vectors

**a**and

**b**are zero, then they are parallel

*A*= |

**a**|(|

**b**|sin

*θ*) = |

**a**×

**b**|

**a**×

**b**is equal to the _______ of the parallelogram determined by

**a**and

**b**.

- volume
- area

*θ*= π / 2, we obtain:

**i**×

**j**=

**k**- −
**k**

**k**

*θ*= π / 2, we obtain:

**j**×

**k**=

- −
**i** **i**

**i**

*θ*= π / 2, we obtain:

**k**×

**i**=

**j**- −
**j**

**j**

*θ*= π / 2, we obtain:

**j**×

**i**=

**k**- −
**k**

**k**

*θ*= π / 2, we obtain:

**k**×

**j**=

**i**- −
**i**

**i**

*θ*= π / 2, we obtain:

**i**×

**k**=

**j**- −
**j**

**j**

*θ*= π / 2, then observe that:

**i**×

**j**≠

**j**×**i****k**

**j**×

**i**

**cross product**is not commutative. Also

**i**× (

**i**×

**j**) =

**j**×**i**= −**k****i**×**k**=**j**

**i**×

**k**=

**j**

**i**×

**i**) ×

**j**= 0 ×

**j**= 0

**cross product**is not commutative. Also

**i**×

**i**) ×

**j**=

- 0 ×
**j**= 0 **j**×**i**= −**k**

**j**= 0

**i**× (

**i**×

**j**) =

**i**×

**k**=

**j**

**a**,

**b**, and

**c**are vectors and

**c**is a scalar, then

**a**×

**b**=

**b**×**a**- −
**b**×**a**

**b**×

**a**

**a**,

**b**, and

**c**are vectors and

**c**is a scalar, then

*c*

**a**) ×

**b**=

*c***b**×*c***a***c*(**a**×**b**) =**a**× (*c***b**)

*c*(

**a**×

**b**) =

**a**× (

*c*

**b**)

**a**,

**b**, and

**c**are vectors and

**c**is a scalar, then

**a**× (

**b**+

**c**) =

**a**×**b**+**a**×**c***c*(**a**×**b**)

**a**×

**b**+

**a**×

**c**

**a**,

**b**, and

**c**are vectors and

**c**is a scalar, then

**a**+

**b**) ×

**c**=

**a**×**c**+**b**×**c****a**×**b**+**a**×**c**

**a**×

**c**+

**b**×

**c**

**a**,

**b**, and

**c**are vectors and

**c**is a scalar, then

**a**• (

**b**×

**c)**=

**a**×**b**×**c**- (
**a**×**b**) •**c**

**a**×

**b**) •

**c**

**a**,

**b**, and

**c**are vectors and

**c**is a scalar, then

**a**× (

**b**×

**c**) =

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

**a**•

**c**)

**b**− (

**a**•

**b**)

**c**

**Scalar Triple Product**

**a**× (**b**×**c**)**a**• (**b**×**c**)**a**× (**b**+**c**)

*V*=

*Ah*= |

**b**×

**c**| |

**a**| |cos

*θ*| =

- |
**a**× (**b**×**c**)| - |
**a**• (**b**×**c**)|

**a**• (

**b**×

**c**)|

*θ*| instead of cos

*θ*in case

*θ*> π / 2 .

**parallelepiped**determined by the vectors

**a**,

**b**, and

**c**is the magnitude of their scalar triple product:

*V*= |**a**× (**b**•**c**)|*V*= |**a**• (**b**×**c**)|

**a**,

**b**, and

**c**is 0, it means that

**a**,

**b**, and

**c**are ________.

- non-coplanar
- coplanar

**Vector Triple Product**

**a**× (**b**×**c**)**a**• (**b**×**c**)**a**× (**b**+**c**)

**a**× (

**b**×

**c**)

**τ**| = |

**r**×

**F**| =

- |
**r**×**F**| sin*θ* - |
**r**×**F**| cos*θ*

**r**×

**F**| sin

*θ*

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