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November 24th, 2017, 06:47 PM   #1
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Learning analytic geometry

Hi, I study alone and I'm having difficulties with these issues, please help me!

Questions:

1- Determine which type of central grid is from its curves.
(x ^ 2) / (a ^ 2) + (y ^ 2) / (b ^ 2) - (z ^ 2) / (c ^ 2) = -1

2- Prove that rotations in R ^ 2 commute.
Ra * Rb = Rb * Ra

3- Prove that roca in R ^ 3 do not commute.
4- How to transform cylindrical coordinates into Cartesian coordinates?
5- ρ sin ϕ = 2
A surface in R ^ 3 is given by coordinate (ρ, θ, ϕ), how to translate into Cartesian coordinates and what is this surface?

Last edited by skipjack; November 24th, 2017 at 09:40 PM.
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November 29th, 2017, 05:11 AM   #2
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Originally Posted by Orgueam View Post
Hi, I study alone and I'm having difficulties with these issues, please help me!
It would help if we knew what kind of difficulties you were having! Could you not just show what you have tried?

Quote:
Questions:

1- Determine which type of central grid is from its curves.
(x ^ 2) / (a ^ 2) + (y ^ 2) / (b ^ 2) - (z ^ 2) / (c ^ 2) = -1
Note that this can be written z^2/c^2= x^2/a^2+ y^2/c^2+ 1

Quote:
2- Prove that rotations in R ^ 2 commute.
Ra * Rb = Rb * Ra
There are different ways to do this. Personally, I would note that any rotation in two dimensions can be written in matrix form $\displaystyle \begin{bmatrix} cos(\theta) & -sin(\theta) \\ sin(\theta) & cos(\theta)\end{bmatrix}$.
Look at $\displaystyle \begin{bmatrix}cos(\theta) & -sin(\theta) \\ sin(\theta) & cos(\theta)\end{bmatrix}\begin{bmatrix}cos(\phi) & -sin(\phi) \\ sin(\phi) & cos(\phi)\end{bmatrix}$ and $\displaystyle \begin{bmatrix}cos(\phi) & -sin(\phi) \\ sin(\phi) & cos(\phi)\end{bmatrix}\begin{bmatrix}cos(\theta) & -sin(\theta) \\ sin(\theta) & cos(\theta)\end{bmatrix}$. Are they the same?

Quote:
3- Prove that roca in R ^ 3 do not commute.
A counter example will suffice. Consider, (1) the rotation about the z- axis by 90 degree and (2) the rotation about the x-axis by 90 degrees. Doing them in the order (1)(2), (1) maps (1, 0, 0) to (0, 1, 0) and then (2) maps (0, 1, 0) to (0, 0, 1). That is, (1)(2) maps (1, 0, 0) to (0, 0, 1). (2) maps (1, 0, 0), since it is on the x-axis to itself, (1, 0, 0), then (1), as before, maps (1, 0, 0) to (0, 1, 0). That is, (2)(1) maps (1, 0, 0) to (0, 1, 0).

Quote:
4- How to transform cylindrical coordinates into Cartesian coordinates?
$\displaystyle r= \sqrt{x^2+ y^2}$, $\displaystyle \theta= arctan\left(\frac{y}{x}\right)$, and z= z. Surely that is given in your mathtbook?

Quote:
5- ρ sin ϕ = 2
A surface in R ^ 3 is given by coordinate (ρ, θ, ϕ), how to translate into Cartesian coordinates and what is this surface?
Spherical coordinates convert to Cartesian coordinates by $\displaystyle x= \rho cos(\theta)sin(\phi)$, $\displaystyle y= \rho sin(\theta)sin(\phi)$, and $\displaystyle z= \rho cos(\phi)$. Squaring the first two and adding, $\displaystyle x^2+ y^2= \rho^2 sin^2(\phi)$ so that, taking the square root of that $\displaystyle \rho sin(\phi)= \sqrt{x^2+ y^2}$. Again, those are basic definitions. Surely they are given in your textbook?
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