I just want to double-check this: if we reverse the order of cross product, even though mathematically we are still getting the same value (which means that we get a parallelepiped that is as large as the one presented on the powerpoint), geometrically, the parallelepiped is not the same one we have here on the powerpoint. That is because the new parallelpiped will end up having 4 points that have different coordinates from those presented.
motoole2
If you reverse the order of cross product, it produces a different value. As mentioned here, the cross product is a noncommutative operation, where u x v = -(v x u). Geometrically, reversing the order for the cross product is kind of like turning the volume inside out, resulting in the volume (and determinant) being negated. The absolute value of the determinant, however, would remain the same and correspond to the volume of this parallelepiped.
loneliboi
I'm trying to understand how all four quantities are the same, and why the first diagram shows volume while others show the area of a specific face of the parallelepiped. How are all of these quantities the same? If the cross product of u and v gives us w. How does w dot w give us volume?
Parker
I was also confused by this, loneliboi. It only made sense to me after looking at the diagram in this URL (https://mathinsight.org/scalar_triple_product). Basically, taking (u x v) dot w as an example, we know that (u x v) gives us the area of the parallelogram formed by the vectors u & v. In other words, the highlighted area of the second image above. From there, we just need to multiply by the height to get the area of the entire parrallelpiped. We know also that the vector given by (u x v) is perpendicular to both u & v. So to get the height, we want the w component of that perpendicular vector. We can get that value via |w|*cos(theta_between_them). All written out, that is evaluates to (u x v) dot w. Please, anyone correct me if I am wrong; this paragraph was as much for my benefit as anyone else's.
I just want to double-check this: if we reverse the order of cross product, even though mathematically we are still getting the same value (which means that we get a parallelepiped that is as large as the one presented on the powerpoint), geometrically, the parallelepiped is not the same one we have here on the powerpoint. That is because the new parallelpiped will end up having 4 points that have different coordinates from those presented.
If you reverse the order of cross product, it produces a different value. As mentioned here, the cross product is a noncommutative operation, where
u x v = -(v x u). Geometrically, reversing the order for the cross product is kind of like turning the volume inside out, resulting in the volume (and determinant) being negated. The absolute value of the determinant, however, would remain the same and correspond to the volume of this parallelepiped.I'm trying to understand how all four quantities are the same, and why the first diagram shows volume while others show the area of a specific face of the parallelepiped. How are all of these quantities the same? If the cross product of u and v gives us w. How does w dot w give us volume?
I was also confused by this, loneliboi. It only made sense to me after looking at the diagram in this URL (https://mathinsight.org/scalar_triple_product). Basically, taking (u x v) dot w as an example, we know that (u x v) gives us the area of the parallelogram formed by the vectors u & v. In other words, the highlighted area of the second image above. From there, we just need to multiply by the height to get the area of the entire parrallelpiped. We know also that the vector given by (u x v) is perpendicular to both u & v. So to get the height, we want the w component of that perpendicular vector. We can get that value via |w|*cos(theta_between_them). All written out, that is evaluates to (u x v) dot w. Please, anyone correct me if I am wrong; this paragraph was as much for my benefit as anyone else's.