How to find the magnitude of this bivector?

[dimension10]

How does one find the value of

$$||A \left(\hat{\imath}\wedge\hat{\jmath}\right) +B(\hat{\imath}\wedge\hat{k})+C( \hat{\jmath}\wedge\hat{k})||$$

Where A, B and C are constant coefficients.


Thanks in advance.

 

[I like Serena]

Hi dimension10!

Assuming your wedges are cross products, I think this section on wiki answers your question:
http://en.wikipedia.org/wiki/Cross_product#Coordinate_notation

 

[micromass]

It could also be that you actually did mean wedges and that you're working with the exterior algebra. In that case, you can define an inner product as follows

$$\langle v\wedge w,v^\prime\wedge w^\prime\rangle= det\left(\begin{array}{cc} \langle v,v^\prime\rangle & \langle v,w^\prime\rangle\\ \langle w,v^\prime\rangle & \langle w,w^\prime\rangle \end{array}\right)$$

Under this inner product, can you now calculate the norm in your example?

See http://en.wikipedia.org/wiki/Exterior_algebra for more details.

 

[dimension10]

It could also be that you actually did mean wedges and that you're working with the exterior algebra. In that case, you can define an inner product as follows

$$\langle v\wedge w,v^\prime\wedge w^\prime\rangle= det\left(\begin{array}{cc} \langle v,v^\prime\rangle & \langle v,w^\prime\rangle\\ \langle w,v^\prime\rangle & \langle w,w^\prime\rangle \end{array}\right)$$

Under this inner product, can you now calculate the norm in your example?

See http://en.wikipedia.org/wiki/Exterior_algebra for more details.

Thanks. However, I don't see any way to express the bivector in that way.

 

[I like Serena]

Thanks. However, I don't see any way to express the bivector in that way.

Ah, okay.
So it's a generalization of the cross product.

Well, it says here:
http://en.wikipedia.org/wiki/Exterior_algebra#Inner_product
that the 3 wedge products you have in your problem form an orthonormal basis.

And I take it the norm would be defined to be the square root of the bivector with itself?

 

[dimension10]

Ah, okay.
So it's a generalization of the cross product.

Well, it says here:
http://en.wikipedia.org/wiki/Exterior_algebra#Inner_product
that the 3 wedge products you have in your problem form an orthonormal basis.

And I take it the norm would be defined to be the square root of the bivector with itself?

But if I were to do that, then I would obtain a 2 by 3 matrix to take the determinant of and determinants are not defined for non-square matrices...

 

[I like Serena]

But if I were to do that, then I would obtain a 2 by 3 matrix to take the determinant of and determinants are not defined for non-square matrices...

How do you get a 2 by 3 matrix?

Note that "orthonormal basis" says it all... no need to take any determinants...

 

[dimension10]

How do you get a 2 by 3 matrix?

Note that "orthonormal basis" says it all... no need to take any determinants...

I don't exactly get how it is an orthonormal basis since A, B, and C are unknown coefficients.

If it were an orthonormal basis, then the magnitude would be 1, but I don't think this will be an orthonormal basis.

 

[I like Serena]

I don't exactly get how it is an orthonormal basis since A, B, and C are unknown coefficients.

If it were an orthonormal basis, then the magnitude would be 1, but I don't think this will be an orthonormal basis.

Yes, A, B, and C are unknown scalar coefficients.

And:
$$\langle \left(\hat{\imath}\wedge\hat{\jmath}\right), \left(\hat{\imath}\wedge\hat{\jmath}\right) \rangle = 1$$$$\langle \left(\hat{\imath}\wedge\hat{\jmath}\right), (\hat{\imath}\wedge\hat{k}) \rangle = 0$$

 

[dimension10]

Yes, A, B, and C are unknown scalar coefficients.

And:
$$\langle \left(\hat{\imath}\wedge\hat{\jmath}\right), \left(\hat{\imath}\wedge\hat{\jmath}\right) \rangle = 1$$$$\langle \left(\hat{\imath}\wedge\hat{\jmath}\right), (\hat{\imath}\wedge\hat{k}) \rangle = 0$$

But to get the bivector into that form, I would need to get the scalars outside the magnitude signs somehow...

 

[I like Serena]

Perhaps you can use the following?
$$||x||=\sqrt{\langle x, x \rangle}$$$$\langle ax+by, z \rangle = a\langle x, z \rangle + b\langle y, z \rangle$$

 

[dimension10]

Perhaps you can use the following?
$$||x||=\sqrt{\langle x, x \rangle}$$$$\langle ax+by, z \rangle = a\langle x, z \rangle + b\langle y, z \rangle$$

Oh, so that is true for bivectors too? I thought it was true only for vectors! Thanks.

 

[I like Serena]

Oh, so that is true for bivectors too? I thought it was true only for vectors! Thanks.

The first is the "usual" definition of the norm.
Since you did not specify you were using another one (and there are many others), this one is assumed.

The second is part of the definition of the inner product, which is the generalized version of the dot product.