What are the results of integrating the dot and cross products of two vectors?

In summary: Then integrate that from 0 to 2.In summary, the dot product of A and B is -8/3 and the cross product is 8/3i + 4j + 16/3k.
  • #1
yusukered07
18
0
If A(t) = t i - t2 j + (t - 1) k and B(t) = 2t2 i + 6t k, evaluate (a) [tex]\int^{2}_{0}A \cdot Bdt ,[/tex] (b) [tex]\int^{2}_{0}A \times B dt.[/tex]
 
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  • #2
I have no idea!

Can you help me out, by SHOWING WHAT YOU HAVE DONE SO FAR?
 
  • #3
yusukered07 said:
1. If A (t) = t i - t2j + (t -1) k, evaluate (b) [tex]\int^{2}_{0} A [/tex]
Integral of t = t2/2 = 2
Integral of t2 = t3/3 = 8/3
Integral of (t-1)=t2/2 - t = 0

Net result 2i -(8/3)j
 
  • #4
mathman said:
Integral of t = t2/2 = 2
Integral of t2 = t3/3 = 8/3
Integral of (t-1)=t2/2 - t = 0

Net result 2i -(8/3)j

Sorry for giving a wrong problem...
 
  • #5
Moderator's note: thread moved from "Calculus & Analysis"

Homework assignments or any textbook style exercises are to be posted in the appropriate forum in our https://www.physicsforums.com/forumdisplay.php?f=152" area. This should be done whether the problem is part of one's assigned coursework or just independent study.


yusukered07 said:
If A(t) = t i - t2 j + (t - 1) k and B(t) = 2t2 i + 6t k, evaluate (a) [tex]\int^{2}_{0}A \cdot Bdt ,[/tex] (b) [tex]\int^{2}_{0}A \times B dt.[/tex]
You need to show an attempt at solving the problem before receiving help!

Start by evaluating A·B and AxB.
 
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  • #6
Redbelly98 said:
Moderator's note: thread moved from "Calculus & Analysis"

Homework assignments or any textbook style exercises are to be posted in the appropriate forum in our https://www.physicsforums.com/forumdisplay.php?f=152" area. This should be done whether the problem is part of one's assigned coursework or just independent study.



You need to show an attempt at solving the problem before receiving help!

Start by evaluating A·B and AxB.

Yeah... That's the process I've made... Then I integrate them with respect to t and evaluate from 0 to 2
 
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  • #7
For the dot product of A and B, I obtained 2*t3+6*t2-6*t. Then integrate that from 0 to 2.

For the cross product, I obtained -6*t3i+(2*t3-8*t2)j+2*t4k.
 

1. What is vector integration?

Vector integration is a mathematical technique used to integrate vector-valued functions. It involves calculating the area under a vector curve, similar to how traditional integration calculates the area under a scalar curve.

2. Why is vector integration important in science?

Vector integration is important in science because many physical quantities, such as velocity and force, are represented by vectors. By integrating these vector quantities, we can determine important properties such as displacement and work.

3. How is vector integration different from traditional integration?

Vector integration differs from traditional integration in that it involves not only calculating the area under a curve, but also taking into account the direction and magnitude of the vector values. This adds an additional level of complexity to the integration process.

4. What are some real-world applications of vector integration?

Vector integration has numerous applications in fields such as physics, engineering, and computer graphics. It can be used to analyze the motion of objects, calculate the work done by a force, and determine the flow of fluids in a system.

5. Are there any specific techniques for solving vector integration problems?

Yes, there are several techniques for solving vector integration problems, including the use of line integrals, surface integrals, and volume integrals. These techniques involve breaking down the integration into smaller, more manageable parts and using specific formulas to solve for the final result.

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