Referring to vectors, determinants, and cross product: as I understand it the determinant of two 2-dimentional vectors yields area, and the determinant of three 3-dimentional vectors gives volume. We used determinants to solve volume/area in the coursework.

So, why are we using cross-product to find volume? Is it simply another method to find the same value?

Thank you

Question

Referring to vectors, determinants, and cross product: as I understand it the determinant of two 2-dimentional vectors yields area, and the determinant of three 3-dimentional vectors gives volume.  We used determinants to solve volume/area in the coursework.

  So, why are we using cross-product to find volume?  Is it simply another method to find the same value?

  Thank you

anonymous · Answer

Correct me if I am wrong. It looks like the magnitude of a cross product actually gives area. So, wouldn't you be using cross products to find area?

I think you are right, determinants of a 2x2 also give area of a parallelogram. So, they will probably be easier to use if your vectors are in 2-space. But what if your vectors lie in 3-space? In that case it may be easier to use the cross-product for area calculation. 

(So cross-products are more general in their application to area calculation. You could use them to find area when your vectors are in 2-space or 3-space.)

anonymous · Answer

Thanks