and let the coordinates of P < sub > 1 </ sub > and P < sub > 2 </ sub > be ( x < sub > 1 </ sub >, y < sub > 1 </ sub >) and ( x < sub > 2 </ sub >, y < sub > 2 </ sub >) respectively.

The three surfaces intersect at the point P ( shown as a black sphere ) with the Cartesian coordinates ( 1, − 1, 1 ).

The spherical coordinates of a point P are then defined as follows:

Let ρ, θ, and φ be spherical coordinates for the source point P. Here θ denotes the angle with the vertical axis, which is contrary to the usual American mathematical notation, but agrees with standard European and physical practice.

The matrix M may be regarded as a diagonal matrix D that has been re-expressed in coordinates of the basis P. In particular, the one-to-one change of variable y = Pz shows that z < sup >*</ sup > Mz is real and positive for any complex vector z if and only if y < sup >*</ sup > Dy is real and positive for any y '; in other words, if D is positive definite.

where x < sub > P </ sub >, y < sub > P </ sub >, and z < sub > P </ sub > are the Cartesian coordinates and i, j and k are the unit vectors along the x, y, and z coordinate axes, respectively.

where the coordinates x < sub > P </ sub >, y < sub > P </ sub >, and z < sub > P </ sub > are each functions of time.

It is often convenient to formulate the trajectory of a particle P ( t ) = ( X ( t ), Y ( t ) and Z ( t )) using polar coordinates in the X-Y plane.

The cylindrical coordinates for P ( t ) can be simplified by introducing the radial and tangential unit vectors,

Now, in general, the trajectory P ( t ) is not constrained to lie on a circular cylinder, so the radius R varies with time, and the trajectory in cylindrical-polor coordinates becomes

It becomes important to then represent the position of particle P in terms of its polar coordinates ( r, θ ).

If P has trilinear coordinates p: q: r, then the vertices L, M, N of the pedal triangle of P are given by

The homothetic center ( which is a triangle center if and only if P is a triangle center ) is the point given in trilinear coordinates by

Say, pick first a point P =( x, y ) with random non-zero coordinates x, y ( mod n ), then pick a random non-zero a ( mod n ), then take b = y < sup > 2 </ sup >-x < sup > 3 </ sup >-ax ( mod n ).

The slope of the tangent line at P is s =( 3x < sup > 2 </ sup >+ 5 )/( 2y )= 4, and then the coordinates of 2P =( x ′, y ′) are x ′= s < sup > 2 </ sup >- 2x = 14 and y ′= s ( x-x ′)- y = 4 ( 1-14 )- 1 =- 53, all numbers understood ( mod n ).

The three coordinates ( ρ, φ, z ) of a point P are defined as:

Let k be an algebraically closed field and let P < sup > n </ sup > be a projective n-space over k. Let f ∈ k ..., x < sub > n </ sub > be a homogeneous polynomial of degree d. It is not well-defined to evaluate f on points in P < sup > n </ sup > in homogeneous coordinates.

In the f-plane the coordinates of the corresponding point are Af.

The data presented are derived almost entirely from X-ray diffraction measurements and include atomic coordinates, cell dimensions, and atomic and ionic radii.

Time coordinates on the TAI scales are conventionally specified using traditional means of specifying days, carried over from non-uniform time standards based on the rotation of the Earth.

Although digital setting circles can be used to display a telescope's RA and Dec coordinates, they are not simply a digital read-out of what can be seen on the telescope's analog setting circles.

These are graphs of ψ ( x, y, z ) functions which depend on the coordinates of one electron.

Alternatively, atomic orbitals refer to functions that depend on the coordinates of one electron ( i. e. orbitals ) but are used as starting points for approximating wave functions that depend on the simultaneous coordinates of all the electrons in an atom or molecule.

The coordinate systems chosen for atomic orbitals are usually spherical coordinates ( r, θ, φ ) in atoms and cartesians ( x, y, z ) in poly-atomic molecules.

In terms of components, these three equations for the conservation of momentum in cylindrical coordinates are

Unlike McCaffrey's black crystal transceivers, Le Guin's ansibles are not mated pairs: it is possible for an ansible's coordinates to be set to any known location of a receiving ansible.

Four points are marked and labeled with their coordinates: ( 2, 3 ) in green, (− 3, 1 ) in red, (− 1. 5 ,− 2. 5 ) in blue, and the origin ( 0, 0 ) in purple.

The coordinates of the points on the curve are of the form ( x, 1 / x ) where x is a number other than 0.

The British Virgin Islands are located in the Caribbean, between the Caribbean Sea and the North Atlantic Ocean, east of Puerto Rico. Its geographic coordinates are.

Its geographical coordinates are.

Informally basis vectors are like " building blocks of a vector ", they are added together to make a vector, and the coordinates are the number of basis vectors in each direction.

The coordinates of the vector are equal to the projections of the vector ( yellow ) onto the x-component basis vector ( green )-using the dot product ( a special case of an inner product, see below ).

respectively, where e < sub > x </ sub >, e < sub > y </ sub >, e < sub > z </ sub > denotes the cartesian basis vectors ( all are orthogonal unit vectors ) and A < sub > x </ sub >, A < sub > y </ sub >, A < sub > z </ sub > are the corresponding coordinates, in the x, y, z directions.

though the coordinates and vectors are now all complex-valued.

The coordinates of the vector are equal to the projections of the vector ( yellow ) onto the x-component basis vector ( green )-using the inner product ( see below ).

Bessel functions are also known as cylinder functions or cylindrical harmonics because they are found in the solution to Laplace's equation in cylindrical coordinates.

To find an object, such as globular cluster NGC 6712, one does not need to look up the RA and Dec coordinates in a book, and then move the telescope to those numerical readings.

If ( x < sub > 1 </ sub >, x < sub > 2 </ sub >, x < sub > 3 </ sub >) are the Cartesian coordinates and ( u < sub > 1 </ sub >, u < sub > 2 </ sub >, u < sub > 3 </ sub >) are the orthogonal coordinates, then

As a consequence, in particular, in the system of coordinates given by the polar representation, the equations then take the form

If ( x, y ) are the Cartesian coordinates of a point, then (− x, y ) are the coordinates of its reflection across the second coordinate axis ( the Y axis ), as if that line were a mirror.

If y is a point where the vector field v ( y ) ≠ 0, then there is a change of coordinates for a region around y where the vector field becomes a series of parallel vectors of the same magnitude.

For the traditional geocache, a geocacher will place a waterproof container containing a log book ( with pen or pencil ) and trade items then record the cache's coordinates.

It is easy enough to " translate " between polar and rectangular coordinates in the plane: let, as above, direction and distance be and respectively, then we have

If the generalized coordinates are represented as a vector and time differentiation is represented by a dot over the variable, then the equations of motion ( known as the Lagrange or Euler – Lagrange equations ) are a set of equations:

However, if I is infinite and the rings R < sub > i </ sub > are non-zero, then the converse is false ; the set of elements with all but finitely many nonzero coordinates forms an ideal which is not a direct product of ideals of the R < sub > i </ sub >.

A product of more than one non-zero rings always has zero divisors: if x is an element of the product all of whose coordinates are zero except p < sub > i </ sub >( x ), and y is an element of the product with all coordinates zero except p < sub > j </ sub >( y ) ( with i ≠ j ), then xy

It can be alternatively stated as: if the span of n vectors has dimension p, then p of these vectors span the space and there is a set of p coordinates on which they are linearly independent.

The concept of spherical coordinates can be extended to higher dimensional spaces and are then referred to as hyperspherical coordinates.

When the coordinates are transformed, for example by rotation or stretching, then the components of the vector also transform.

For example, if v consists of the x, y, and z-components of velocity, then v is a contravariant vector: if the coordinates of space are stretched, rotated, or twisted, then the components of the velocity transform in the same way.

The solution in cylindrical coordinates was then extended to the optical regime by Joseph B. Keller, who introduced the notion of diffraction coefficients through his geometrical theory of diffraction ( GTD ).

Each element is created and manipulated numerically ; essentially using Cartesian coordinates for the placement of key points, and then a mathematical algorithm to connect the dots and define the colors.

Going to polar coordinates then yields the integral for a circular object of radius a ( see for example Born and Wolf ):

This approach allows neurosurgeons to obtain a number of coronal images, which are then used to calculate the stereotactic coordinates of the place in the anterior cingulate cortex, where lesions need to be created.