1. Coordinate Systems

a. Rectangular Coordinates - 2D, 3D and nD

3. Rectangular Coordinates in Higher Dimensions (Optional)

The ideas of \(2\) dimensional space (a plane, \(\mathbb{R}^2\)) and \(3\) dimensional space (\(\mathbb{R}^3\)) can be generalized to any number of dimensions although we cannot actually visualize it. A point \(P\) in \(n\)-dimensional space is specified by giving an ordered \(n\) -tuple such as \(P=(p_1,p_2,\cdots,p_n)\). Since it takes \(n\) \(\mathbb{R}\)eal numbers to specify a point, \(n\)-dimensional space is also called \(\mathbb{R}^n\). The origin is the point \((0,0,\cdots,0)\) and there are \(n\) axes, one for each coordinate, \(x_i\). The \(x_i\)-axis is the set of points of the form \((0,0,\cdots,x_i,\cdots,0)\).

In \(\mathbb{R}^4\), we take the coordinates as \((x,y,z,w)\). We think of \(x\), \(y\) and \(z\) as horizontal and \(w\) as vertical. The figure shows a hypercube in \(\mathbb{R}^4\) whose base is a cube (shown in red) in the horizontal \(xyz\)-hyperplane.

4D Rectangular Coordinates — 4D Rectangular Solid
by Avek Dongol

1. Coordinate Systems

a. Rectangular Coordinates - 2D, 3D and nD

3. Rectangular Coordinates in Higher Dimensions (Optional)

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