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Two lines, vectors, planes, etc., are said to be perpendicular if they meet at a right angle. In R^n, two vectors a and b are perpendicular if their dot product a·b=0. (1) In ...

An abstract vector space of dimension n over a field k is the set of all formal expressions a_1v_1+a_2v_2+...+a_nv_n, (1) where {v_1,v_2,...,v_n} is a given set of n objects ...

Let a particle travel a distance s(t) as a function of time t (here, s can be thought of as the arc length of the curve traced out by the particle). The speed (the scalar ...

A Lie algebra is a vector space g with a Lie bracket [X,Y], satisfying the Jacobi identity. Hence any element X gives a linear transformation given by ad(X)(Y)=[X,Y], (1) ...

In a given circle, find an isosceles triangle whose legs pass through two given points inside the circle. This can be restated as: from two points in the plane of a circle, ...

Let M^n be a compact n-dimensional oriented Riemannian manifold without boundary, let O be a group representation of pi_1(M) by orthogonal matrices, and let E(O) be the ...

The anticomplement of a point P in a reference triangle DeltaABC is a point P^' satisfying the vector equation P^'G^->=2GP^->, (1) where G is the triangle centroid of ...

A square matrix A is antihermitian if it satisfies A^(H)=-A, (1) where A^(H) is the adjoint. For example, the matrix [i 1+i 2i; -1+i 5i 3; 2i -3 0] (2) is an antihermitian ...

Arc length is defined as the length along a curve, s=int_gamma|dl|, (1) where dl is a differential displacement vector along a curve gamma. For example, for a circle of ...

An augmented matrix is a matrix obtained by adjoining a row or column vector, or sometimes another matrix with the same vertical dimension. The most common use of an ...