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

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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

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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.)
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