rockyshephear said:
Why is the Cross Product needed in the first place? As a simplification, I would think that if one is presented with two vectors and asked to find the parallelogram formed, it's much simpler to just multiply the two magnitudes and have done with it. Just like in non-vector mathematics.
"Have done with it" and just ignore the fact that it gives the
wrong answer? The area of a parallelogram is the product of the two magnitudes only if the parallelogram happens to be a rectangle. The area of parallelogram is |u||v| sin(\theta) where \theta is the angle between the sides, exactly the formula for magnitude of the cross product of two vectors.
And why would the vector have to be normal to the plane that contains the two vectors? Is that arbitrary?
I answered that before. Distinguishing the normal to a surface is an important ability in applications and that is an important use of cross product. If a surface is given in terms of two parameters, u and v, with "position vector", \vec{r}(u,v), then the derivatives, \vec{r}_u and \vec{r}_v, are two vectors in the tangent plane to the surface. The "fundamental vector product", \vec{r}_u\times\vec{r}_v is a vector normal to the surface. It's magnitude, \left|\vec{r}_u\times\vec{r}_v\right| times "dudv", \left|\vec{r}_u\times\vec{r}_v\right|dudv is the "differential of surface area". To integrate a real-valued function over the surface, you write it as \int\int f(u,v)\left|\vec{r}_u\times\vec{r}_v\right| dudv. Even simpler is integrating an
vector valued function over a surface (as for example, finding the flow through a surface or the flux of a magnetic field through the surface). That is just \int\int \vec{f}(u,v)\cdot\vec{r}_u\times\vec{r}_v dudv
Can I now create a new vector calculus rule called the ROCKY PRODUCT that defines a paralellogram fromed by A and B in a plane, but the pseudovector is at 45 degrees to the plane holding the vectors. Or an infinite amount of new rules at differing angles?
Anyone see what I'm driving at?
Thanks
Of course you can! Is it
useful? I can think of many situations in which it is useful to separate a vector into components along and perpendicular to a surface: only the component of force
perpendicular to a surface causes pressure on the surface, only the component of force
along a surface will move an object over the surface. Can you give examples where a component at 45 degrees to a surface or at some other specific angle would be useful?
Apropos your post on "torque", you don't see to understand that torque is always measured relative to some
axis. And is always perpendicular to that axis. Perhaps instead of railing against things you don't understand, you should
learn them.
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