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In physics, a vector field (,,) is a function that returns a vector and is defined for each point (with coordinates ,,) in a region of space. The idea of sources and sinks applies to b {\displaystyle \mathbf {b} } if it follows a continuity equation of the form
Illustration of the vector formulation. The equation of a line can be given in vector form: = + Here a is the position of a point on the line, and n is a unit vector in the direction of the line. Then as scalar t varies, x gives the locus of the line.
A free vector is a vector quantity having an undefined support or region of application; it can be freely translated with no consequences; a displacement vector is a prototypical example of free vector. Aside from the notion of units and support, physical vector quantities may also differ from Euclidean vectors in terms of metric.
A unit vector is any vector with a length of one; normally unit vectors are used simply to indicate direction. A vector of arbitrary length can be divided by its length to create a unit vector. [14] This is known as normalizing a vector. A unit vector is often indicated with a hat as in â.
The vector equation for a line is = + where is a unit vector in the direction of the line, is a point on the line, and is a scalar in the real number domain. Substituting the equation for the line into the equation for the plane gives
: origin of the line : distance from the origin of the line : direction of line (a non-zero vector) Searching for points that are on the line and on the sphere means combining the equations and solving for , involving the dot product of vectors:
Distance from the origin O to the line E calculated with the Hesse normal form. Normal vector in red, line in green, point O shown in blue. In analytic geometry, the Hesse normal form (named after Otto Hesse) is an equation used to describe a line in the Euclidean plane, a plane in Euclidean space, or a hyperplane in higher dimensions.
Isotropic quadratic form A quadratic form q is said to be isotropic if there is a non-zero vector v such that q(v) = 0; such a v is an isotropic vector or null vector. In complex geometry, a line through the origin in the direction of an isotropic vector is an isotropic line. Isotropic coordinates