![]() If you find this linear transformation business too abstract to help you appreciate the utility of the scalar triple product, you could also think about triple integrals, whose definition is based on chopping up a region into small boxes. In particular, the above determinant form of the scalar triple product is key, as matrices are strongly linked to linear transformations. Here comes the scalar triple product, as it measures the volume that is changing. One of these properties is how linear maps expand or contract objects. Thus, properties of linear maps can be seen by how they transform parallelepipeds. It turns out that three-dimensional linear transformations always map parallelepipeds into other parallelepipeds. ![]() The route to parallelepipeds comes through these linear functions, which we'll call linear transformations or linear maps to emphasize how they map objects into other objects. Bottom line of this: linear functions are fundamental in calculus. Calculus is all about the infinitesimal (i.e., making everything small), so the small structure you see when zooming in is fundamental. In a nutshell, differentiability means that a function looks linear if you zoom in. The reason stems from the definition of the differentiability of functions. In multivariable calculus, it turns out there are parallelepipeds lurking behind some important formulas and theorems. Nonetheless, the scalar triple product does have its uses even if you aren't that excited about parallelepipeds. Its applications are more immediate, and its use is more widespread. If you have only enough available brain cells to master either the cross product or the scalar triple product, we'd recommend focusing on the cross product. To begin with, we recommend you first master the cross product. But, if you don't happen to find yourself pining to know the volume of a parallelepiped, you may wonder what's the use of the scalar triple product. The scalar triple product is obviously very useful if you have a lot of parallelepipeds lying around and want to know their volume. Parallelepiped using the scalar triple product. In case you like to see it with numbers, here's an example of calculating the volume of a If you keep the figure rotating by dragging it with the mouse, you'll see it much better. The three-dimensional perspective of this graph is hard to perceive when the graph is still. The scalar triple product of three vectors $\vc$ is shown by the red vector its magnitude is the area of the highlighted parallelogram, which is one face of the parallelepiped.
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