- #1

- 23

- 0

Take for example the dot or scalar product.

Two nice, and perfectly reasonable, ways to define the dot product in 2 dimensional space are as follows.

Let A=(a1, a2) and B=(b1,b2). Let us also define the angle these two vectors make to be ω

1) A.B=|A||B|cos(ω)

2) A.B=a1*b1+a2*b2

Now, if we accept one of these definitions as 'law' then it should be possible to deduce the other definition and vice versa.

Let us take 2) as our definition.

Then it must be possible to deduce |A||B|cos(ω)=a1*b1+a2*b2.

I have tried doing this by setting |A|=(a1^2+a2^2)^0.5 but then you get stuck with cos(ω) and I cannot see a way of relating cos(ω) with the components of A and B without using the definition I'm trying to deduce!

Both definitions of the dot product are used happily in n dimensional space. I find this quite unsettling. It is possible to geometrically prove the 1) and 2) are equal in 2D and possible, albeit slightly harder, to also prove equality in 3D. When we extend this theorem to 4D there is no way of proving equality so how is it that it is used so freely in n-dimensional space! There isn't even a concept for cos(ω) for 4d and beyond. Does anyone else have these issues or is it me just not understanding something fundamental?

I have another question that is kind of unrelated. Rather than posting a new thread I shall just make an unusually large post.

Elipses, hyperbolas and parabolas are all cases of conic sections. Taking a plane and intersecting it with a cone. The different possibilities of this intersection gives rise to these different curves. My question is, taking the definition of these curves to be slices through a cone how does one deduce the general equations of the curves? It must be possible to deduce, for example, the equation of an ellipse from this method. It must also give rise to the eccentricity relationship between the shapes (i.e. a circle has eccentricity of 0). A relationship which I have never truly undestood. I have tried to do this and have failed to make any real progress. I'd appreciate it if someone gave me a helping hand!

I apologise for the essay ^^