Integration Involving Dot Product?

In summary: Of course, you could also have\int \vec{A}\times\vec{B} d\vec{r}but that would require knowing the specific parameterization of the vectors.
  • #1
FeDeX_LaTeX
Gold Member
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Hello;

How do I solve an integral involving the dot product?

For example, imagine I have two vectors A and B, and I want to calculate;

[tex]\int \vec{A}\cdot\vec{B}[/tex]

How do I do this?

I am asking because, I read somewhere that I have to evaluate the dot product first:

[tex]\int |A||B|\cos\theta[/tex]

But I have no idea where to go from here. I know integrating [tex]\cos\theta[/tex] gives me [tex]\sin\theta[/tex], but don't know what to do about the two magnitudes of vectors A and B.

Thanks
 
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  • #2
Hello,

Your question seems ill-posed. Are [itex] \vec A, \vec B [/itex] constant vectors? I assume that they must be functions. If so, how are they parameterized?

As for the second part of your question, if you're are lucky and the vectors are parameterized in such a way that they have constant magnitude, then |A| and |B| are just constants. However, in this case it is likely the relative angle will be a complicated expression.
 
  • #3
FeDeX_LaTeX said:
Hello;

How do I solve an integral involving the dot product?

For example, imagine I have two vectors A and B, and I want to calculate;

[tex]\int \vec{A}\cdot\vec{B}[/tex]

How do I do this?

I am asking because, I read somewhere that I have to evaluate the dot product first:

[tex]\int |A||B|\cos\theta[/tex]

But I have no idea where to go from here. I know integrating [tex]\cos\theta[/tex] gives me [tex]\sin\theta[/tex], but don't know what to do about the two magnitudes of vectors A and B.
You have omitted the differential in your integral, so it's impossible to say what the antiderivative is. You would integrate cos(theta) as you showed only if integration was being done with respect to theta. That is, if the differential was [itex]d\theta[/itex].
 
  • #4
The dot product is a number so the integral of [itex]\vec{A}\cdot\vec{B}[/itex] is just the integral of that numerical function, no different from what you learned in Calculus I. To integrate the dot product of the vector functions [itex]\vec{A}= 2x\vec{i}+ 3\vec{j}+ e^x\vec{k}[/itex] and [itex]\vec{B}= x^2\vec{i}+ x^2\vec{j}+ 2\vec{k}[/itex] you take their dot product, [itex](2x)(x^2)+ 3(x^2)+ e^x(2)= 2x^3+ 3x^2+ 2e^x[/itex] and integrate that:
[tex]\int \vec{A}\cdot\vec{B}dx= \int 2x^3+ 3x^2+ 2e^x dx[/tex]

As others have pointed out, your failure to write "dx" or "dy" or whatever the variable of integration is makes it impossible to specify more. You should NOT use "[itex]\vec{A}\cdot\vec{B}= |\vec{A}||\vec{B}|cos(\theta)[/itex]" unless you know how the angle between the vectors, [itex]\theta[/itex], and the lengths of the vectors [itex]|\vec{A}|[/itex] and [itex]\vec{B}[/itex] vary with what ever the variable of integration.

Perhaps if you were to give a specific integral, we could say more.
 
  • #5
Thanks, I understand. Sorry for omitting the differential.

Do I do the same thing for the cross product?
 
  • #6
That's a bit more complicated since the cross product of two vectors is a vector, not a number. In that case you could have either:
[tex]\int \vec{A}\times\vec{B}\cdot d\vec{S}[/tex]
which would be a "standard" numerical integral or you could have
[tex]\int \vec{A}\times\vec{B} dx[/itex]
which would indicate a "component by component" integration and would yield a vector.
 

What is the dot product?

The dot product, also known as the scalar product, is a mathematical operation that takes two vectors as input and returns a single scalar value. It is calculated by multiplying the magnitudes of the two vectors and the cosine of the angle between them.

What is integration involving dot product?

Integration involving dot product is a way to calculate the area under the curve of a dot product function. It involves finding the integral of the dot product of two vectors over a given interval.

Why is integration involving dot product useful?

Integration involving dot product is useful in many areas of science and engineering, particularly in physics and mathematics. It allows us to calculate the work done by a force, the displacement of an object, and the change in energy of a system.

What are some common applications of integration involving dot product?

Integration involving dot product is commonly used in mechanics, electromagnetism, and quantum mechanics. It is also used in fields such as computer graphics, robotics, and signal processing.

What are some techniques for solving integration involving dot product?

Some common techniques for solving integration involving dot product include using trigonometric identities, substitution, and integration by parts. It is also helpful to have a strong understanding of vector operations and calculus principles.

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