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Their cross product is a normal vector to that plane, and any vector orthogonal to this cross product through the initial point will lie in the plane. [1] This leads to the following coplanarity test using a scalar triple product: Four distinct points, x 1, x 2, x 3, x 4, are coplanar if and only if,
In physics, specifically classical mechanics, the three-body problem is to take the initial positions and velocities (or momenta) of three point masses that orbit each other in space and calculate their subsequent trajectories using Newton's laws of motion and Newton's law of universal gravitation. [1]
In physics, Lami's theorem is an equation relating the magnitudes of three coplanar, concurrent and non-collinear vectors, which keeps an object in static equilibrium, with the angles directly opposite to the corresponding vectors. According to the theorem,
(M2) at most dimension 1 if it has no more than 1 line, (M3) at most dimension 2 if it has no more than 1 plane, and so on. It is a general theorem (a consequence of axiom (3)) that all coplanar lines intersect—the very principle that projective geometry was originally intended to embody.
[1] Parallel lines are the subject of Euclid's parallel postulate. [2] Parallelism is primarily a property of affine geometries and Euclidean geometry is a special instance of this type of geometry. In some other geometries, such as hyperbolic geometry, lines can have analogous properties that are referred to as parallelism.
A rotation symmetry in dimension 2 or 3 must move a lattice point to a succession of other lattice points in the same plane, generating a regular polygon of coplanar lattice points. We now confine our attention to the plane in which the symmetry acts ( Scherrer 1946 ), illustrated with lattice vectors in the figure.
The two important examples are (i) the internal forces in a rigid body, and (ii) the constraint forces at an ideal joint. Lanczos [1] presents this as the postulate: "The virtual work of the forces of reaction is always zero for any virtual displacement which is in harmony with the given kinematic constraints." The argument is as follows.
[1] The law of reflection states that a reflected ray of light emerges from the reflecting surface at the same angle to the surface normal as the incident ray, but on the opposing side of the surface normal in the plane formed by the incident and reflected rays. This behavior was first described by Hero of Alexandria (AD c. 10–70). [2]