Integration Involving Dot Product?

[FeDeX_LaTeX]

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;

$$\int \vec{A}\cdot\vec{B}$$

How do I do this?

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

$$\int |A||B|\cos\theta$$

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

Thanks

 

[Kreizhn]

Hello,

Your question seems ill-posed. Are $\vec A, \vec B$ 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.

 

[Mark44]

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;

$$\int \vec{A}\cdot\vec{B}$$

How do I do this?

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

$$\int |A||B|\cos\theta$$

But I have no idea where to go from here. I know integrating $$\cos\theta$$ gives me $$\sin\theta$$, 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 $d\theta$.

 

[HallsofIvy]

The dot product is a number so the integral of $\vec{A}\cdot\vec{B}$ 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 $\vec{A}= 2x\vec{i}+ 3\vec{j}+ e^x\vec{k}$ and $\vec{B}= x^2\vec{i}+ x^2\vec{j}+ 2\vec{k}$ you take their dot product, $(2x)(x^2)+ 3(x^2)+ e^x(2)= 2x^3+ 3x^2+ 2e^x$ and integrate that:
$$\int \vec{A}\cdot\vec{B}dx= \int 2x^3+ 3x^2+ 2e^x dx$$

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 "$\vec{A}\cdot\vec{B}= |\vec{A}||\vec{B}|cos(\theta)$" unless you know how the angle between the vectors, $\theta$, and the lengths of the vectors $|\vec{A}|$ and $\vec{B}$ vary with what ever the variable of integration.

Perhaps if you were to give a specific integral, we could say more.

 

[FeDeX_LaTeX]

Thanks, I understand. Sorry for omitting the differential.

Do I do the same thing for the cross product?

 

[HallsofIvy]

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:
$$\int \vec{A}\times\vec{B}\cdot d\vec{S}$$
which would be a "standard" numerical integral or you could have
[tex]\int \vec{A}\times\vec{B} dx[/itex] <br /> which would indicate a "component by component" integration and would yield a vector.[/tex]