Get started today!

Good to have you back!
If you've signed in to StudyBlue with Facebook in the past, please do that again.

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

Jason S.

The **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

vector

If **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}〉

Advertisement
)

2

3

(**a** × **b**)

One of the most important properties of the **cross product** is given by the following theorem.

The vector **a **× **b** is __________ to both **a** and **b**.

- orthogonal
- parallel

orthogonal

(**a** × **b**) • **a** = 0

(**a** × **b**) • **b** = 0

Therefore **a** × **b** is orthogonal to both __ and __

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

To find the length of |**a** × **b**|

If *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*

To find the length of |**a** • **b**|

If *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*

Two nonzero vectors **a** and **b** are parallel if and only if

**a**×**b**= 1**a**×**b**= 0

If the **cross product **of two nonzero vectors **a** and **b** are zero, then they are parallel

The length of the cross product **a** × **b** is equal to the _______ of the parallelogram determined by **a** and **b**.

- volume
- area

area

If *θ* = π / 2, we obtain:

**k**- −
**k**

If *θ* = π / 2, we obtain:**j** × **k** =

- −
**i** **i**

If *θ* = π / 2, we obtain:**k** × **i** =

**j**- −
**j**

Advertisement

If *θ* = π / 2, we obtain:**j **×** i **=

**k**- −
**k**

−**k**

If *θ* = π / 2, we obtain:**k** × **j** =

**i**- −
**i**

−**i**

If *θ* = π / 2, we obtain:**i** × **k** =

**j**- −
**j**

−**j**

If *θ* = π / 2, then observe that:**i** × **j** ≠

**j**×**i****k**

Thus the **cross product** is not commutative. Also** i **× (**i** × **j**) =

**j**×**i**= −**k****i**×**k**=**j**

Also Observe

(**i** × **i**) × **j** = 0 × **j** = 0

Thus the **cross product** is not commutative. Also

(**i** × **i**) × **j** =

- 0 ×
**j**= 0 **j**×**i**= −**k**

0 × **j** = 0

**i** × (**i** × **j**) = **i** × **k** = **j**

Also Observe

If **a**, **b**, and **c** are vectors and **c** is a scalar, then**a** × **b** =

**b**×**a**- −
**b**×**a**

−**b** × **a**

If **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**)

If **a**, **b**, and **c** are vectors and **c** is a scalar, then**a** × (**b** + **c**) =

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

If **a**, **b**, and **c** are vectors and **c** is a scalar, then

(**a** + **b**) × **c** =

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

If **a**, **b**, and **c** are vectors and **c** is a scalar, then**a** • (**b** × **c)** =

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

(**a** × **b**) • **c**

If **a**, **b**, and **c** are vectors and **c** is a scalar, then

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

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

Which is the **Scalar Triple Product**

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

The volume of the parallelepiped is *V* = *Ah* = |**b** × **c**| |**a**| |cos *θ*| =

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

|**a** • (**b** × **c**)|

We must use |cos *θ*| instead of cos *θ* in case *θ* > π / 2 .

The volume of the **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**)|

V = |a • (b × c)|

When the volume of a parallelepiped determined by **a**, **b**, and **c** is 0, it means that **a**, **b**, and **c** are ________.

- non-coplanar
- coplanar

coplanar

The vectors must lie in the same plane

Which is the **Vector Triple Product**

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

Used to derive Kepler’s First Law of planetary motion

Torque

|**τ**| = |**r** × **F**| =

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

|**r** × **F**| sin *θ*

* The material on this site is created by StudyBlue users. StudyBlue is not affiliated with, sponsored by or endorsed by the academic institution or instructor.

"StudyBlue is great for studying. I love the study guides, flashcards and quizzes. So extremely helpful for all of my classes!"

Alice , Arizona State University"I'm a student using StudyBlue, and I can 100% say that it helps me so much. Study materials for almost every subject in school are available in StudyBlue. It is so helpful for my education!"

Tim , University of Florida"StudyBlue provides way more features than other studying apps, and thus allows me to learn very quickly!??I actually feel much more comfortable taking my exams after I study with this app. It's amazing!"

Jennifer , Rutgers University"I love flashcards but carrying around physical flashcards is cumbersome and simply outdated. StudyBlue is exactly what I was looking for!"

Justin , LSU