- Age: 24
- Sex: straight single man
Recent Status: A vector, as defined below, is a specific mathematical structure.
About Me: Example: (2,-5), (-1, 0, 2), (4.5), and (PI, a, b, 2/3) are all examples of vectors of dimension 2, 3, 1, and 4 respectively. The first vector has components 2 and -5. Note: Alternately, an "unordered" collection of n elements {A1, A2, ..., An} is called a "set." Definition: Two vectors are equal if their corresponding components are equal. Example: If A = (-2, 1) and B = (-2, 1), then A = B since -2 = -2 and 1 = 1. However, (5, 3) not_equal (3, 5) because even though they have the same components, 3 and 5, the component do not occur in the same order. Contrast this with sets, where {5, 3} = {3, 5}. Definition: The magnitude of a vector A of dimension n, denoted |A|, is defined as |A| = sqrt(A1^2 + A2^2 + ... + An^2) Geometrically speaking, magnitude is synonymous with "length," "distance", or "speed." In the two-dimensional case, the point represented by the vector A = (A1, A2) has a distance from the origin (0, 0) of sqrt(A1^2 + A2^2) according to the pythagorean theorem. In the three-dimension case, the point represented by the vector A = (A1, A2, A3) has a distance from the origin of sqrt(A1^2 + A2^2 + A3^2) according to the three-dimensional form of the Pythagorean theorem (A box with sides a, b, and c has a diagonal of length sqrt(a2+b2+c2) ). With vectors of dimension n greater than three, our geometric intuition fails, but the algebraic definition remains. Definition: The sum of two vectors A = (A1, A2, ..., An) and B = (B1, B2, ..., Bn) is defined as A + B = (A1 + B1, A2 + B2, ..., An + Bn) Note: Addition of vectors is only defined if both vectors have the same dimension. Example: (2, -3) + (0, 1) = (2+0, -3+1) = (2, -2). (0.1, 2) + (-1, PI) = (0.1 + -1, 2 + PI) = (-0.9, 2+PI)
