Triple cross product proof
Keywords: triple cross product proof
Description: Definition of the scalar triple product and derivation of its formula. Include interactive graphic to illustrate its properties.
The scalar triple product of three vectors $\vc$, $\vc$, and $\vc
This formula for the volume can be understood from the above figure. The volume of the parallelepiped is the area of the base times the height. From the geometric definition of the cross product. we know that its magnitude, $\|\vc \times \vc\|$, is the area of the parallelogram base, and that the direction of the vector $\vc \times \vc$ is perpendicular to the base. The height of the parallelepiped is the component of $\vc
|\cos \phi |$, where $\phi$ is the angle between $\vc
|\cos \phi | = |(\vc \times \vc) \cdot \vc
The below applet can help you understand the properties of the scalar triple product $(\vc \times \vc) \cdot \vc
The scalar triple product can be positive, negative, or zero. (That's why we need the absolute value for the volume.) What determines the sign of $(\vc \times \vc) \cdot \vc
Scalar triple product. The value of the scalar triple product $(\color
The three-dimensional perspective of this graph is hard to perceive when the graph is still. If you keep the figure rotating by dragging it with the mouse, you'll see it much better.
In case you like to see it with numbers, here's an example of calculating the volume of a parallelepiped using 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. 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.
To begin with, we recommend you first master the cross product. 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. Its applications are more immediate, and its use is more widespread.
Nonetheless, the scalar triple product does have its uses even if you aren't that excited about parallelepipeds. In multivariable calculus, it turns out there are parallelepipeds lurking behind some important formulas and theorems. The reason stems from the definition of the differentiability of functions. In a nutshell, differentiability means that a function looks linear if you zoom in. Calculus is all about the infinitesimal (i.e. making everything small), so the small structure you see when zooming in is fundamental. Bottom line of this: linear functions are fundamental in calculus.
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. It turns out that three-dimensional linear transformations always map parallelepipeds into other parallelepipeds. Thus, properties of linear maps can be seen by how they transform parallelepipeds. One of these properties is how linear maps expand or contract objects. Here comes the scalar triple product, as it measures the volume that is changing. In particular, the above determinant form of the scalar triple product is key, as matrices are strongly linked to linear transformations .
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. If you are thinking that a box is a type of parallelepiped and small means a linear transformation is relevant, then you definitely need to get out more and do other things besides think about math. But those connections are exactly how the scalar triple product gets tied to triple integrals. We don't need a scalar triple product for a regular triple integral, though, as we know how to calculate the volume of a box without it. But, when you start changing variables in triple integrals. then the box gets transformed into a parallelepiped, and the scalar triple product volume calculation becomes important.