origin

Examples of origin in the following topics:

• The Cross Product

  • The cross product of two vectors is a vector which is perpendicular to both of the original vectors.
  • The result is a vector which is perpendicular to both of the original vectors.
  • Because it is perpendicular to both original vectors, the resulting vector is normal to the plane of the original vectors.
  • If the two original vectors are parallel to each other, the cross product will be zero.
  • The magnitude of vector $c$ is equal to the area of the parallelogram made by the two original vectors.

• Curve Sketching

  • If the sum of the degrees of $x$ and $y$ in each term is always even or always odd, then the curve is symmetric about the origin and the origin is called a center of the curve.
  • If the curve passes through the origin then determine the tangent lines there.

• Cylindrical and Spherical Coordinates

  • A spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the radial distance of that point from a fixed origin, its polar angle measured from a fixed zenith direction, and the azimuth angle of its orthogonal projection on a reference plane that passes through the origin and is orthogonal to the zenith, measured from a fixed reference direction on that plane.
  • Spherical coordinates are useful in connection with objects and phenomena that have spherical symmetry, such as an electric charge located at the origin.
  • A cylindrical coordinate system with origin $O$, polar axis $A$, and longitudinal axis $L$.

• Vectors in Three Dimensions

  • A bound vector is determined by the coordinates of the terminal point, its initial point always having the coordinates of the origin $O = (0,0,0)$.
  • Thus the bound vector represented by $(1,0,0)$ is a vector of unit length pointing from the origin along the positive $x$-axis.

• Models Using Differential Equations

  • The mathematical theory of differential equations first developed together with the sciences where the equations had originated and where the results found application.
  • However, diverse problems, sometimes originating in quite distinct scientific fields, may give rise to identical differential equations.

• Limits and Continuity

  • For example, the function $f(x,y) = \frac{x^2y}{x^4+y^2}$ approaches zero along any line through the origin.
  • However, when the origin is approached along a parabola $y = x^2$, it has a limit of $0.5$.

• Derivatives of Trigonometric Functions

  • In the context of the standard unit circle with radius 1, where a triangle is formed by a ray originating at the origin and making some angle with the $x$-axis, the sine of the angle gives the length of the $y$-component (rise) of the triangle, the cosine gives the length of the $x$-component (run), and the tangent function gives the slope ($y$-component divided by the $x$-component).

• Solving Differential Equations

  • Solving this equation shows that $f(x)$ is equal to the negative of its derivative; therefore, the function $f(x)$ must be $e^{-x}$, as the derivative of this function equals the negative of the original function.
  • A complete solution contains the same number of arbitrary constants as the order of the original equation.

• Three-Dimensional Coordinate Systems

  • Also known as analytical geometry, this system is used to describe every point in three dimensional space in three parameters, each perpendicular to the other two at the origin.

• The Divergence Theorem

  • Let's consider a system with a point charge $Q$ located at the origin.
  • We will apply the divergence theorem for a sphere of radius $R$, whose center is also at the origin.