Vector Products: What Happens When Professors Refuse to Explain?

In summary, the conversation discusses why a vector product is not generally defined component-wise, and the concept of pointwise is also brought up. The speaker's professor has mentioned that this will become clear later, but the conversation continues to question the use and definition of a vector product. The speaker suggests a possible definition, but it is ultimately deemed useless at the level being discussed. Instead, scalar and vector products are typically defined based on magnitudes and angles, and do not correspond to the operation mentioned earlier.
  • #1
Treadstone 71
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Why is a vector product not generally defined pointwise? My professor simply gave a mysterious "you'll see why later". What's the worst that can happen?
 
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  • #2
What do you mean 'pointwise'? And for that matter what do you mean by 'a vector product', there is generally considered to be one such. Presumably he means something like 'because there is no nice formula for what v^w is in terms of v and w', though I'm not sure I agree with that. Maybe if you even gave what your definition is we might have some better idea.
 
  • #3
By "pointwise" I mean "component-wise". Define x:VXV->V as (a,b,c,d)x(e,f,g,h)=(ae,bf,cg,dh) to be the vector multiplication on some vector space V.
 
  • #4
Oh, right, so that's what you mean. Since vector product has a strict meaning wondered what you were up to, as does pointwise which is strictly different from what you've done: it literally is used to talk about things happening at one point at a time.

Well you can define a 'multiplication' like that, it is certainly a bilinear map from VxV to V that does a lot of interseting things, it even has a name that I can't recall (when applied to a vector space of matrices), and it makes V into an algebra. But it is however, at the level you're looking at useless: it does nothing geometric, or nice.
 
  • #5
As matt grime has pointed out, there is at most levels no need to introduce this kind of a product.

Instead we have scalar and vector products defined (on [itex]\mathbb{R}^3[/itex]) as the products of magnitudes, the cosine/sine of the angle between them (and we make the vector product into a vector by the right-hand screw rule).

Then we can work out what to do with the components once we've got a definite basis. And neither of these correspond to the operation you've defined.
 

1. What are "Vector Products" and why are professors refusing to explain them?

"Vector Products" refer to products that involve the use of vectors, which are mathematical quantities with both magnitude and direction. Professors may refuse to explain these products due to the complexity of vector mathematics or because they expect students to have a basic understanding of vectors already.

2. How can I learn about Vector Products if my professor refuses to explain them?

If your professor refuses to explain Vector Products, you can try finding other resources such as textbooks, online tutorials, or seeking help from classmates or tutors. It may also be helpful to review basic vector concepts and mathematical operations.

3. Are Vector Products important to understand in the field of science?

Yes, Vector Products are crucial in many scientific fields such as physics, engineering, and computer science. They are used to describe and analyze various physical phenomena and are also essential in problem-solving and modeling.

4. What are some common applications of Vector Products?

Some common applications of Vector Products include calculating forces and velocities in physics problems, determining the direction and magnitude of electric and magnetic fields, and creating computer graphics and animations.

5. Are there any tips for understanding Vector Products more easily?

Some tips for understanding Vector Products include reviewing basic vector concepts and operations, practicing with different types of vector problems, and seeking help from resources such as textbooks, online tutorials, and tutors. It may also be helpful to visualize vectors using diagrams or drawings.

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