I am not currently in school, but need to relearn some vector algebra for an electrodynamics class. I have a textbook that I am using to teach myself the basics.
Here's the problem, in one section on the scalar triple product (A · (B x C)), it states that geometrically the magnitude of this (|A · (B x C)|) represents the volume of the parallelpiped generated by the 3 vectors A, B and C since |B x C| is the area of the base and |A cos ϴ| is the height.
I get what they are saying, but mathematically it does not work out that way. Assume A = (1x + 0y + 1z) and B = (0x + 1y + 1z). The length (magnitude) of both A and B is √2. So the area of the base formed by these vectors would be 2; however, A x B = √3.
Can someone explain where I am going wrong? Or is the book wrong?
The area of the base is indeed a cross product's modulus. Your mistake is in multiplying vectors' lengths as if the base is a rectangle. However, the angle between sides of a parallelogram clearly matters. As a sanity check, note that with an extremely small angle there should be next to no area. As for why the cross product works, note the true height is a side multiplied by a sine (draw a diagram if it helps you see why). When $\vec{A},\,\vec{B}$ are angles separated by an angle $\theta$, $|\vec{A}\times\vec{B}|=|\vec{A}||\vec{B}||\sin\theta|$.