Two planes intersect at an angle strictly between 0° and 90°. One of them contains a hyperbola, which we project onto the other intersecting plane. Is the result still a hyperbola ? And if the answer is affirmative, does this mean that the answer to this question is also a `yes`, inasmuch as any triangle can be seen as the projection of an equilateral one onto a plane at a given angle ? — 79.113.235.46 ( talk) 06:13, 17 October 2016 (UTC) reply

If one of the axes of the hyperbola is parallel (or normal) to the line of intersection of the two planes: yes (by a change of variables in the conic equation). I suspect it's true in only that case ( an example of an obviously non-hyperbolic projection of a hyperbola), and so is not useful for the triangle question. -- Tardis ( talk) 00:54, 18 October 2016 (UTC) reply

Let ${\displaystyle I=\{0,1,\dots ,2^{n}-1\}}$, ${\displaystyle A=\{|y_{i}\rangle =\sum _{x_{i}\in I}{(-1)^{a_{i}}|x_{i}\rangle }\mid a_{i}\in I\}}$, ${\displaystyle f(|y_{i}\rangle )=\oplus a_{i}}$. (the XOR of the phases' exponents). We say that ${\displaystyle |x\rangle =|y\rangle }$ iff ${\displaystyle \forall i:|x_{i}|^{2}=|y_{i}|^{2}}$ (two vectors are equivalent iff all of their amplitudes are equal).

Is there any unitary transformation ${\displaystyle U}$ that satisfies: ${\displaystyle \forall |a\rangle ,|b\rangle \in A:f(|a\rangle )\neq f(|b\rangle )\implies U|a\rangle \not \equiv U|b\rangle }$?

Thanks in advance! עברית ( talk) —Preceding undated comment added 06:27, 17 October 2016 (UTC) reply

• The definition of A sounds fishy, but in any case, a prominent notice at the top of the page states: We don't do your homework for you, though we’ll help you past the stuck point. Tigraan ^{Click here to contact me} 11:09, 17 October 2016 (UTC) reply

Let a be a positive real number. What is the measure of the angle formed by the graph of ${\displaystyle f(x)=a|x|}$? GeoffreyT2000 ( talk, contribs) 16:03, 17 October 2016 (UTC) reply

• Just compute the angle between ${\displaystyle f(x)=ax}$ and the vertical line, and double that. Are you familiar with tangent (trigonometry)? Tigraan ^{Click here to contact me} 17:04, 17 October 2016 (UTC) reply
  • The angle formed between ${\displaystyle f(x)=ax}$ and the vertical line has the same value as the angle formed between ${\displaystyle f(x)=ax}$ and the horizontal line. Logic dictates that it must be half the angle formed between the horizontal line and the vertical line. 175.45.116.99 ( talk) 04:48, 19 October 2016 (UTC) reply
    • The IP's answer is only true for a = 1. However, for a > 0, the angle between y = ax and the x-axis is tan⁻¹a, the requested angle is 2(π / 2 – tan⁻¹a) = π – 2tan⁻¹a. As expected, as a → 0⁺, the angle → π. As a → +∞, angle → 0, and at a = 1, angle = π / 2. If you wanted an angle of π / 3, solve for a to get a = √3. EdChem ( talk) 05:25, 19 October 2016 (UTC) reply

      ... and, of course, all of the above assumes identical scales on the axes. Dbfirs 15:54, 19 October 2016 (UTC) reply