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• Cauchy-Schwarz Inequality
  ...ion shows that <math>\mathbf{p}</math> and <math>\mathbf{q}</math> form an orthogonal basis of the linear subspace spanned by <math>\mathbf{u}</math> and <math>\ ...s established that <math>\mathbf{p}</math> and <math>\mathbf{q}</math> are orthogonal, and that the sum of their squared norms is <math>1</math>. Now we have
  13 KB (2,048 words) - 15:28, 22 February 2024
• 2003 AIME II Problems/Problem 11
  Let <math>N</math> be the orthogonal projection from <math>C</math> to <math>AB.</math> Thus, <math>[CDM]=\frac{
  5 KB (772 words) - 19:47, 1 August 2023
• Finite Fourier Analysis
  ...ove that the set of eigenvectors {<math>{f_n : n=0,\ldots, N-1}</math>} is orthogonal, and that the norm of <math>f_n</math> is <math>\sqrt{N}</math>. We can no
  4 KB (724 words) - 19:15, 9 September 2006
• 2007 USAMO Problems/Problem 6
  ...>, <math>F</math>. Since <math>IE\perp AE</math>, <math>\omega</math> is [[orthogonal]] to the inversion circle, so <math>\mathcal{I}(\omega)=\omega</math>. Cons
  7 KB (1,274 words) - 15:11, 31 August 2017
• 2007 Cyprus MO/Lyceum/Problems
  The volume of an orthogonal parallelepiped is <math>132\;\mathrm{cm}^3 </math> and its dimensions are i In the following figure <math>ABCD</math> is an orthogonal trapezium with <math>\ang A= \ang D=90^\circ</math> and bases <math>AB = a<
  13 KB (1,990 words) - 08:29, 19 December 2009
• 2007 Cyprus MO/Lyceum/Problem 10
  The volume of an orthogonal [[parallelepiped]] is <math>132\;\mathrm{cm}^3 </math> and its dimensions a
  969 bytes (139 words) - 16:47, 6 May 2007
• 2007 Cyprus MO/Lyceum/Problem 22
  In the figure, <math>ABCD</math> is an orthogonal trapezium with <math>\angle A= \angle D=90^\circ</math> and bases <math>AB
  979 bytes (166 words) - 02:33, 19 January 2024
• De Longchamps point
  ...rthogonal to the <math>\omega_C.</math> Prove that de Longchamps circle is orthogonal to the <math>\omega_A, \omega_B, A-</math>power, <math>B-</math>power and <
  10 KB (1,780 words) - 09:23, 17 November 2022
• 1995 AHSME Problems/Problem 30
  ...32,\frac 32\right)</math>. The plane that passes through this point and is orthogonal to the diagonal has the equation <math>x+y+z=\frac 92</math>.
  5 KB (828 words) - 05:52, 30 September 2023
• 2007 AMC 12B Problems/Problem 25
  ...n, the k unit vector must be zero. Also, because the i unit vector must be orthogonal and also 0. Thus, the vector of line <math>DE</math> is simply <math>2tj+2k
  3 KB (470 words) - 19:46, 17 July 2023
• 2002 USAMO Problems/Problem 6
  ...tial blocks (5 oriented the same way, including itself, and 9 oriented the orthogonal way). Therefore <math>14b(n)\geq 2n(n - 2)</math> or
  5 KB (940 words) - 17:33, 16 July 2014
• 2004 AMC 12B Problems/Problem 6
  The directions "southwest" and "southeast" are orthogonal. Thus the described situation is a right triangle with legs <math>8</math>
  1 KB (154 words) - 19:23, 22 October 2022
• 1999 AHSME Problems/Problem 22
  Each of the graphs consists of two orthogonal half-lines. In the first graph both point downwards at a <math>45^\circ</ma
  3 KB (538 words) - 12:02, 17 October 2020
• 2009 AMC 12B Problems/Problem 22
  ...>B'=(s,0)</math>, <math>C'=(s+t,0)</math> and <math>D'=(t,0)</math> be the orthogonal projections of <math>B</math>, <math>C</math>, and <math>D</math> onto the
  6 KB (1,071 words) - 22:25, 9 October 2021
• 2009 AMC 10B Problems/Problem 22
  As the line <math>RN</math> is orthogonal to <math>QM</math>, it must have the equation <math>y+2x+q=0</math> for som ...r. This rotation will map the segment <math>MQ</math> to a segment that is orthogonal to <math>MQ</math>, contains <math>R</math> and contains the midpoint of <m
  12 KB (1,868 words) - 03:36, 30 September 2023
• 2011 AIME I Problems/Problem 13
  ...e <math>BD</math> makes a <math>2-14-10\sqrt{2}</math> right triangle, the orthogonal line makes the same right triangle rotated <math>90^\circ</math>. Therefore
  4 KB (613 words) - 11:37, 8 September 2022
• 2013 USAJMO Problems/Problem 5
  ...cyclic, and triangle <math>AXQ</math> is rectangle, and <math>CX</math> is orthogonal to <math>AZ</math>. Now
  7 KB (1,250 words) - 18:05, 1 October 2021
• 2014 AIME II Problems/Problem 14
  ...</math>. But <math>MO</math> and <math>AH</math> are parallel, both being orthogonal to <math>BC</math>. Therefore <math>MH:AO=MC:CO</math>, or <math>MH=\dfrac
  11 KB (1,442 words) - 19:28, 21 October 2023
• 1972 IMO Problems/Problem 6
  ...r of their <math>z</math>-coordinates. First take a plane <math>\pi</math> orthogonal to <math>\pi_i</math>, which cuts <math>\pi_1,\pi_2,\pi_3</math> along thre
  7 KB (1,370 words) - 15:42, 29 January 2021
• Asymptote: Crash Course
  So now that we've figured out exactly what the heck are perspective and orthogonal projection, let's find out how to do it in Asymptote.
  3 KB (410 words) - 20:22, 15 December 2020
• 2006 iTest Problems
  ...eral triangle <math>ABC</math>. <math>D</math> and <math>E</math> are the orthogonal projections of <math>B</math> and <math>C</math> onto <math>\ell</math> res
  31 KB (4,811 words) - 00:02, 4 November 2023
• 2017 AIME II Problems/Problem 15
  ...n if <math>X</math> is any point inside tetrahedron <math>ABCD</math>, its orthogonal projection onto line <math>MN</math> will have smaller <math>f</math>-value
  6 KB (971 words) - 02:08, 22 January 2024
• 2018 AMC 12B Problems/Problem 23
  Let <math>D</math> be the orthogonal projection of <math>B</math> onto the equator. Note that <math>\angle BDA = Let <math>D</math> be the orthogonal projection of <math>B</math> onto the equator. Note that <math>\angle BDA =
  6 KB (984 words) - 22:39, 3 December 2021
• 2018 IMO Problems/Problem 6
  The circle <math>BDF</math> is orthogonal to the circle <math>\theta</math> <i><b>(Claim 2).</b></i> The circles <math>BDF</math> and <math>BDE</math> are orthogonal to the circle <math>\omega</math> <i><b>(Claim 2).</b></i>
  8 KB (1,407 words) - 01:47, 19 November 2023
• 2006 iTest Problems/Problem 27
  ...eral triangle <math>ABC</math>. <math>D</math> and <math>E</math> are the orthogonal projections of <math>B</math> and <math>C</math> onto <math>\ell</math> res
  3 KB (488 words) - 12:54, 7 December 2018
• 2017 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 8
  ...ength <math>r</math> in the unit direction <math>(12,5)/13</math> which is orthogonal to the line AB.
  2 KB (278 words) - 04:37, 19 January 2019
• 2020 AMC 10A Problems/Problem 20
  ...rrow{\mathbf{BC}}</math> and <math>\overrightarrow{\mathbf{BA}}</math> are orthogonal, and their dot-product is <math>0</math>.
  13 KB (2,046 words) - 18:33, 28 October 2023
• 2021 AIME I Problems/Problem 13
  ...{2}r</math>. Let <math>x=O_{1}O_{2}</math> and <math>O^{\prime}</math> the orthogonal projection of <math>O</math> onto line <math>\ell</math>. Define the functi
  17 KB (2,852 words) - 03:59, 7 February 2024
• 2021 AMC 12A Problems/Problem 20
  ...math>\ell</math>, and further let <math>X</math> and <math>Y</math> be the orthogonal projections of <math>F</math> and <math>V</math> onto <math>AQ</math>. Beca
  4 KB (796 words) - 17:33, 13 July 2021
• 1979 IMO Problems/Problem 4
  ...</math> on the plane <math>p</math>. Then, the line <math>PT</math> is the orthogonal projection of the line <math>PQ</math> on the plane <math>p</math>, and thu
  4 KB (662 words) - 23:01, 29 January 2021
• 2023 AIME I Problems/Problem 14
  ...- a + 1,\ j + a)</math>. We observe the equivalent result for steps in the orthogonal direction.
  11 KB (1,834 words) - 22:01, 4 January 2024
• 2023 AIME II Problems/Problem 12
  (Note that in our problem, since <math>AP</math> and <math>BC</math> are not orthogonal, (<math>ABC</math> isn't isosceles) this is enough to show that <math>BQCP<
  12 KB (1,900 words) - 18:14, 28 January 2024
• Orthonormal
  ...t of "orthonormal" refers to the fact that distinct vectors in the set are orthogonal (perpendicular): their [[dot product]] is zero. The "normal" part refers to
  3 KB (518 words) - 22:17, 26 June 2023
• 1992 IMO Problems/Problem 5
  ...math>,<math>S_{y}</math>,<math>S_{z}</math>, be the sets consisting of the orthogonal projections of the points of <math>S</math> onto the <math>yz</math>-plane, ...otes the number of elements in the finite set <math>|A|</math>. (Note: The orthogonal projection of a point onto a plane is the foot of the perpendicular from th
  3 KB (560 words) - 00:43, 17 November 2023
• 1990 OIM Problems/Problem 4
  b. Prove that there is a circle orthogonal to all the circles <math>C_2</math>. NOTE: Two circles are orthogonal if they intersect and the respective tangents at the points of intersection
  862 bytes (152 words) - 13:41, 13 December 2023
• 1995 OIM Problems/Problem 3
  Let <math>r</math> and <math>s</math> be two orthogonal lines that are not in the same plane. Let <math>AB</math> be their common p
  767 bytes (136 words) - 14:48, 13 December 2023