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Cross product
- Intro Lesson: a13:34
- Intro Lesson: b11:27
- Intro Lesson: c4:01
- Lesson: 15:55
- Lesson: 24:52
- Lesson: 37:47
- Lesson: 410:32
- Lesson: 511:46
Cross product
Lessons
Notes:
Cross Product
Suppose given two vectors A and B, you want to find a third vector that is perpendicular to them. To find the third vector, we need to do the cross product.
Let A=<a1,a2,a3> and B=<b1,b2,b3>. Then the cross product of these two vectors will be:
Cross Product is NOT commutative
Keep in mind that
Let u,v and w be vectors and c is a scalar. Then we have the following cross product properties:
Cross Product
Suppose given two vectors A and B, you want to find a third vector that is perpendicular to them. To find the third vector, we need to do the cross product.
Let A=<a1,a2,a3> and B=<b1,b2,b3>. Then the cross product of these two vectors will be:
A×B=<a2b3−b2a3,b1a3−a1b3,a1b2−a2b1>
The formula is ugly to remember, so we have a technique that we will show you in the introduction videos!Cross Product is NOT commutative
Keep in mind that
A×B≠B×A
However, the relationship between these two cross products is that they are pointing in different directions. Hence,A×B=−(B×A)
Properties of Cross ProductsLet u,v and w be vectors and c is a scalar. Then we have the following cross product properties:
- u×(v+w)=u×v+u×w
- (u+v)×w=u×w+v×w
- (cu)×v=c(u×v)=u×(cv)
- u⋅(v×w)=(u×v)⋅w
- u×(v×w)=(u⋅w)v−(u⋅v)w
- IntroductionCross Product Overviewa)Cross Product
- A vector perpendicular to the other two
- Formula for the cross product
- Technique to use cross product
b)Cross Product is Not Commutative- Order in which you do cross product matters!
- A×B≠B×A
- A×B=−(B×A)
c)Properties of Cross Product- Algebraic properties
- Distribution property
- Scalar multiplication property
- Etc.
- 1.Using the Cross Product
If u=<3,−2,4> and v=<−1,2,−5>, then compute u×v. - 2.If u=<0,1,−3> and v=<1,0,4>, then compute u×v.
- 3.Showing Cross Product is Not Commutative Let A=<1,0,3> and B=<−2,1,0>. Show that:
A×B≠B×A
- 4.Verifying Properties of Cross Product Suppose c=2, and u=<2,−1,4> and v=<4,2,1>. Verify that
(cu)×v=c(u×v)
- 5.Suppose u=<2,−1,4>, v=<4,2,1> and w=<1,−1,2>. Verify that
u×(v+w)=u×v+u×w