Clear up some cross product confusion for me

In summary, we need to use the distributive law and the fact that the cross product of two parallel vectors is 0 to find the cross product of A and B, which is (x2 - y2)r0Xr1.
  • #1
2slowtogofast
135
1
Here is what I have

A=xro+yr1

B=yro+xr1


I need to find A x B

I am confused about do it because The components of A and B are in terms of vectors

If A = 3i + 4j + 7k and B = 2i + 2j + 1k I would have no problem (these numbers are meaningless just giving an example) finding A x B but I don't have components I have vectors. Can some one explain how to do this. Thanks
 
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  • #2
Could I set it up as follows

Sorry I don't no how to input a matrix hopefully this can get the idea across
ro r1 k ro r1
x y 0 x y
y x 0 y x

and then do the normal cross product thing where you multiply diagonals and so on. The thing that troubles me is there is really no k component. The only thing I know is my answer will be perpindicular to the plane with r0 and r1 so not sure what to do
 
  • #3
AXB = (x2 - y2)r0Xr1.

I am assuming x and y are scalars.
 
  • #4
hi 2slowtogofast! :smile:
2slowtogofast said:
Here is what I have

A=xro+yr1

B=yro+xr1


I need to find A x B

hints: distributive law

what are ro x ro and r1 x r1 ? :wink:
 
  • #5
@tinytim

roxro = 0 I can see how that will simplify things I am confused about the distributive law part

A x (B+D) = (AxB)+(AxD)

I have somthing more like this

(xro+yr1) x (yro+xr1) = (xro+yr1) x yro + (xro+yr1) x xr1

Use the distributive rule again on the RHS

(xro x yro) + (yr1 x yro) + (xro x xr1) + (yr1 x r1)

Then the first and last terms are 0. Am on the right track or did I make a wrong turn somewhere
 
  • #6
hi 2slowtogofast! :smile:

(just got up :zzz:)
2slowtogofast said:
(xro x yro) + (yr1 x yro) + (xro x xr1) + (yr1 x r1)

Then the first and last terms are 0. Am on the right track

yes :smile:

now what is the relationship between r1 x ro and ro x r1 ? :wink:
 
  • #7
r0 x r1 = -(r1 x r0)

If that is correct i think i can take it from there thanks for the help
 
  • #8
2slowtogofast said:
r0 x r1 = -(r1 x r0)

that's it! :smile:
 

1. What is a cross product?

A cross product is a mathematical operation that takes two vectors and produces a new vector that is perpendicular to both of the original vectors. It is denoted by the symbol "x" or "⨯".

2. How is a cross product different from a dot product?

While a dot product results in a scalar (a single number), a cross product results in a vector. The dot product measures the similarity between two vectors, while the cross product measures the perpendicularity.

3. What are the applications of cross product in real life?

Cross product has many applications in physics, engineering, and computer graphics. It is used to calculate torque, magnetic fields, and angular momentum. In computer graphics, it is used to determine the orientation of 3D objects and to create realistic lighting effects.

4. How do you calculate a cross product?

To calculate the cross product of two 3D vectors, you can use the formula:

a x b = (a2b3 - a3b2, a3b1 - a1b3, a1b2 - a2b1)

where a and b are the two vectors. This formula can also be represented using determinants.

5. Can a cross product be negative?

Yes, a cross product can be negative. The direction of the resulting vector is determined by the right-hand rule, where the direction of the cross product is perpendicular to both vectors and follows the direction of a right-handed screw. If the vectors are in the opposite direction, the resulting vector will be negative.

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