Vectors

[Prashantkrishnan]

In 3D, with bases e1, e2, e3, we get the cross product of two vectors a <a1, a2, a3> and b <b1, b2, b3> as the following determinant:

Code: Select all

e1   e2   e3
          a1   a2   a3
          b1   b2   b3

This gives the area of the parallelogram formed by a and b as a vector. It's magnitude is ab sin t where t is the angle between a and b.

So the cross product in 4D, with bases e1, e2, e3, e4, becomes a ternary operation giving the volume of a parallelepiped as a vector.

In 3D, 1D length and 2D area can both be described as vectors. 3D volumes cannot be described as vectors because we cannot associate a unique direction with a volume.

In 4D, 1D length and 3D volume can be expressed as vectors while 4D bulk cannot. 2D areas cannot be associated with a unique direction, but any plane can be specified by giving two non collinear vectors on it. So how do we express 2D area in 4D space? Definitely not as a vector, but is it possible to use a tensor of rank 2?

[wendy]

If you write the vectors as columns of a matrix, the 'determinate' of the matrix gives the volume of it, in any dimension. In number theory, this is used to determine the primes that divide a vector.

The representation of the cross product is the determinate of the root vectors (x, y, z), and the two other vectors expressed therein. In four dimensions, one supposes the bivector might arise from the product of two vector-units (w,x,y,z), and the remaining two vectors.

[wendy]

The interesting thing in higere space, is that an m-dimensional space, holds a normal n-dimensional space in m+n dimensions, and that the sense of the space can be transferred. For example a 2D surface can transfer its in/out nature to a 1D vector in 3D, but in 4D, a 2D fabric can not support an in/out relation, but rather a circulation. [The notion here is that orthogonal 2-spaces can be part of a clifford-parallel set, which transfers direction as well as orientation].

The space orthogonal to upwards and forward is across. The across-space is N-2 dimensions, but we can always transfer the notion of parity here. If we suppose that we have a room, and forward is into the 4th dimension, and up is our up, we can lay clocks on the floor, it is not possible to cause the 12's to point in the same direction, but the clocks will still point upwards when the numbers are clockwise in the floor.

[PatrickPowers]

In 3D, with bases e1, e2, e3, we get the cross product of two vectors a <a1, a2, a3> and b <b1, b2, b3> as the following determinant:

Code: Select all

e1   e2   e3
          a1   a2   a3
          b1   b2   b3

This gives the area of the parallelogram formed by a and b as a vector. It's magnitude is ab sin t where t is the angle between a and b.

So the cross product in 4D, with bases e1, e2, e3, e4, becomes a ternary operation giving the volume of a parallelepiped as a vector.

In 3D, 1D length and 2D area can both be described as vectors. 3D volumes cannot be described as vectors because we cannot associate a unique direction with a volume.

In 4D, 1D length and 3D volume can be expressed as vectors while 4D bulk cannot. 2D areas cannot be associated with a unique direction, but any plane can be specified by giving two non collinear vectors on it. So how do we express 2D area in 4D space? Definitely not as a vector, but is it possible to use a tensor of rank 2?

This is perfect for geometric algrebra. In that formalism 2D spaces are expressed as bivectors, an ordered pair of vectors and a scalar. You can think of the scalar as the area, but it has other uses as well. For 3D spaces there are trivectors and so forth.

Geometric algebra is the same no matter the number of dimensions. It even works in hyperbolic spaces. You can define subspaces of any dimension with ease. No more of those weird kludges that only work in 3D.