The Fourth Dimension

Cover The Fourth Dimension
The Fourth Dimension
Edward Hope Neville
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The spaces with the equations ^'^R + f^'VR+ ^' ^R + '^''Tr + k =0, are parallel if and only if the vectors (X', fu, ', v', trr'), (X", /x", v", -sr") are collinear.
A plane has a pair of equations, the equations of a pair of spaces that include it, but there is no one pair of equations that can be regarded as canonical. Any two linear equations taken together re- present a plane unless the spaces which they represent individually are parallel.
For a line three equations are required, and three
...linear equations together represent a line unless two of them represent parallel spaces or represent spaces whose plane of intersection is parallel to the space represented by the third equation.
Ex. Xxix. The space through (3, - 1, 5, 1) with the vecspace discussed in Example xv (p. 19) has the equation 9(^-3) + 34(r;+l) + 41(C-5)+18(T-l) = 0, that is, 9^ + 347; + 41^ + 18r -216 = 0.
Ex. XXX. The intersection of a line with a space is calculated most readily if the line is given parametrically and the space by an equation.


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