Testwiki:Reference desk/Archives/Mathematics/2016 October 17

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October 17

Plane Projection of a Hyperbola

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)

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)

Unitary Transformation

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

Is there any unitary transformation $U$ that satisfies: $\mathrm{\forall }|a⟩,|b⟩\in A:f(|a⟩)\ne f(|b⟩)⟹U|a⟩\equiv ̸U|b⟩$?

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

• The definition of A sounds fishy, but in any case, a prominent notice at the top of the page states: Template:Tq Tigraan^{Click here to contact me} 11:09, 17 October 2016 (UTC)

Measure of angle formed by absolute value graph

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

• Just compute the angle between $f(x)=ax$ 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)
  • The angle formed between $f(x)=ax$ and the vertical line has the same value as the angle formed between $f(x)=ax$ 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)
    • 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)

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