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• Cauchy-Schwarz Inequality
  ...t <math>\overrightarrow{v} = t \overrightarrow{w}</math>, or if one of the vectors is zero. This formulation comes in handy in linear algebra problems at inte Consider the vectors <math> \mathbf{a} = \langle a_1, \ldots a_n \rangle </math> and <math> {} \
  13 KB (2,048 words) - 15:28, 22 February 2024
• Vector
  ...gle brackets or parentheses, <math>(x\,\,y\,\,z\,\,...)</math>. The set of vectors over a [[field]] is called a [[vector space]]. ...enient to consider a vector as originating from the origin. This way, two vectors can be compared by only looking at their endpoints. This is why we only req
  11 KB (1,875 words) - 15:35, 7 January 2025
• Rational approximation
  ...<math>(\{kx_1\},\dots,\{kx_m\})</math>, <math>k=0,\dots, n^m</math> in the unit cube <math>[0,1)^m</math> divided into <math>n^m</math> equal subcubes.
  7 KB (1,290 words) - 12:18, 30 May 2019
• 1984 AIME Problems/Problem 9
  P=s*markscalefactor*unit(A-B)+B; R=s*markscalefactor*unit(C-B)+B;
  6 KB (947 words) - 18:00, 20 February 2025
• Volume
  The volume of a [[parallelepiped]] spanned by vectors <math>\bold{a} = a_1\bold{i} + a_2\bold{j} + a_3\bold{k}, \bold{b} = b_1\bo *[[Center]]s of adjacent [[face]]s of a unit [[cube (geometry) | cube]] are joined to form a regular [[octahedron]]. Wha
  3 KB (523 words) - 20:24, 17 August 2023
• Euler line
  ...circumcenter be represented by the vector <math>O = (0, 0)</math>, and let vectors <math>A,B,C</math> correspond to the vertices of the triangle. It is well k ...<math>\vec O = \vec X(3)</math> as origin and <math>\vec {HO}</math> as a unit vector.
  59 KB (10,203 words) - 04:47, 30 August 2023
• Fermat point
  ...ec{A},b=\vec{B},c=\vec{C},x=\vec{X}</math> and <math>i,j,k</math> the unit vectors along <math>a,b,c</math>. Then, <math>|a|=a\cdot i=(a-x)\cdot i+x\cdot i\le ...on, it is easy to see the point is the point <math>P</math> where the unit vectors to the vertices sum to <math>0</math>. For a quadrilateral, it is the inter
  4 KB (769 words) - 16:07, 29 December 2019
• 2007 AMC 12B Problems/Problem 25
  ...nd <math>CD</math>. Finding this cross product, we take the determinant of vectors ...th>z</math> direction, 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</ma
  3 KB (470 words) - 19:46, 17 July 2023
• 2010 AMC 12B Problems/Problem 18
  ...the orthocenter of the triangle whose vertices are the heads of the three vectors. Also, it's well known that the orthocenter of a triangle lies inside its c
  6 KB (1,105 words) - 13:39, 9 January 2024
• 1973 IMO Problems
  ...P_1}, \overrightarrow{OP_2}, \ldots, \overrightarrow{OP_n}</math> are unit vectors such that points <math>P_1, P_2, \ldots, P_n</math> all lie in a plane cont
  3 KB (602 words) - 15:44, 29 January 2021
• 2011 AIME I Problems/Problem 3
  The unit basis vectors of our new axes are <math>$$\left(\begin{array}{c}12/13\\ 5/13\end{array}\right)$$</math
  4 KB (699 words) - 16:01, 13 April 2024
• 2011 USAMO Problems/Problem 3
  ...th>AB=DE=p</math>, <math>BC=EF=q</math>, <math>CD=FA=r</math>. Define the vectors: <cmath>\vec{u} = \vec{AB} + \vec{DE}</cmath> <cmath>\vec{v} = \vec{BC} + \ ...vec{w}</math>. As <math>\angle MNX = \gamma</math>, the angle between the vectors <math>\vec{u}</math> and <math>\vec{NE}</math> is <math>\gamma</math>. Sim
  11 KB (1,925 words) - 12:07, 31 August 2023
• 2012 AMC 12B Problems/Problem 23
  ...th> must be algebraic. Since <math>z_0</math> is algebraic and lies on the unit circle, <math>z_0</math> must be a root of unity (Comment: this is not true However, by inspection <math>4x^4+4x^3+4x^2+4x+4</math> has roots on the unit circle, because <math>x^4+x^3+x^2+x+1=(x^5-1)/(x-1)</math> which brings the
  11 KB (1,979 words) - 17:25, 6 September 2021
• 1979 USAMO Problems/Problem 2
  ...nd <math>C</math> are all unit vectors because all those points are on the unit sphere. ...<math>N</math>, <math>A</math>, <math>B</math>, and <math>C</math> as unit vectors with their coordinates.
  6 KB (1,013 words) - 22:09, 21 November 2023
• 1971 IMO Problems/Problem 5
  ...h>, there are exactly <math>m</math> points in <math>S</math> which are at unit distance from <math>A</math>. I shall prove a more general statement about the unit distance graph(<math>V=\mathbb{R}^2</math>, adjacency iff the Euclidean dis
  19 KB (3,570 words) - 01:18, 13 January 2025
• 1972 IMO Problems/Problem 6
  ...{v}_p</math> perpendicular to all of these planes (there are two such unit vectors, and we have to choose one of them), and then, by the directed distance bet
  7 KB (1,370 words) - 15:42, 29 January 2021
• 2014 AIME I Problems/Problem 11
  ...th>-coordinate grid and then makes a sequence of six moves. Each move is 1 unit in a direction parallel to one of the coordinate axes. Each move is selecte ...e that a movement up, right, left, or down in the old coordinates adds the vectors <math>\langle 1, -1 \rangle</math>, <math>\langle 1, 1 \rangle</math>, <mat
  11 KB (1,677 words) - 23:54, 4 February 2022
• 2018 AMC 12A Problems/Problem 23
  == Solution 8 (Vectors) == ...the average of any two unit vectors is average of the arguments of the two vectors. Thereby, the acute angle formed is <cmath>\frac{36^\circ{} + 180^\circ{} -
  12 KB (1,878 words) - 00:53, 2 September 2024
• 1973 IMO Problems/Problem 1
  ...OP_1}, \overrightarrow{OP_2},\cdots, \overrightarrow{OP_n}</math> are unit vectors such that points <math>P_1, P_2, \cdots, P_n</math> all lie in a plane cont ...<math>\ge 1</math> betwen two with norm <math>1</math>. The sum of the two vectors of norm <math>1</math> makes an angle of <math>\le\frac\pi 2</math> with th
  1 KB (227 words) - 15:45, 29 January 2021
• 2018 Putnam B Problems
  Let <math>\mathcal{P}</math> be the set of vectors defined by ...s + z^{n-1}</math>. Prove that <math>f_n</math> has no roots in the closed unit disk <math>\{z \in \mathbb{C}: |z| \le 1\}</math>.
  2 KB (369 words) - 23:35, 21 April 2019

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