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

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October 17

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Plane Projection of a Hyperbola

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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]

Unitary Transformation

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Let  ,  ,  . (the XOR of the phases' exponents). We say that   iff   (two vectors are equivalent iff all of their amplitudes are equal).

Is there any unitary transformation   that satisfies:  ?

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. TigraanClick here to contact me 11:09, 17 October 2016 (UTC)[reply]

Measure of angle formed by absolute value graph

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Let a be a positive real number. What is the measure of the angle formed by the graph of  ? GeoffreyT2000 (talk, contribs) 16:03, 17 October 2016 (UTC)[reply]

  • Just compute the angle between   and the vertical line, and double that. Are you familiar with tangent (trigonometry)? TigraanClick here to contact me 17:04, 17 October 2016 (UTC)[reply]
    • The angle formed between   and the vertical line has the same value as the angle formed between   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]