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The vector projection (also known as the vector component or vector resolution) of a vector a on (or onto) a nonzero vector b is the orthogonal projection of a onto a straight line parallel to b. The projection of a onto b is often written as proj b a {\displaystyle \operatorname {proj} _{\mathbf {b} }\mathbf {a} } or a ∥ b .
A square matrix is called a projection matrix if it is equal to its square, i.e. if =. [2]: p. 38 A square matrix is called an orthogonal projection matrix if = = for a real matrix, and respectively = = for a complex matrix, where denotes the transpose of and denotes the adjoint or Hermitian transpose of .
The vector projection of a vector on a nonzero vector is defined as [note 1] = , , , where , denotes the inner product of the vectors and . This means that proj u ( v ) {\displaystyle \operatorname {proj} _{\mathbf {u} }(\mathbf {v} )} is the orthogonal projection of v {\displaystyle \mathbf {v} } onto the line spanned by u ...
Also, let Q = (x 1, y 1) be any point on this line and n the vector (a, b) starting at point Q. The vector n is perpendicular to the line, and the distance d from point P to the line is equal to the length of the orthogonal projection of on n. The length of this projection is given by:
ρ is the length of the vector projected onto the xy-plane, φ is the angle between the projection of the vector onto the xy-plane (i.e. ρ) and the positive x-axis (0 ≤ φ < 2π), z is the regular z-coordinate. (ρ, φ, z) is given in Cartesian coordinates by:
The Leray projection, named after Jean Leray, is a linear operator used in the theory of partial differential equations, specifically in the fields of fluid dynamics. Informally, it can be seen as the projection on the divergence-free vector fields.
Projectivization is a special case of the factorization by a group action: the projective space P(V) is the quotient of the open set V \ {0} of nonzero vectors by the action of the multiplicative group of the base field by scalar transformations. The dimension of P(V) in the sense of algebraic geometry is one less than the dimension of the ...
In mathematics, a projection is an idempotent mapping of a set (or other mathematical structure) into a subset (or sub-structure). In this case, idempotent means that projecting twice is the same as projecting once. The restriction to a subspace of a projection is also called a projection, even if the idempotence property is lost. An everyday ...