Resource: Perfect triples, “nice” unit vectors, and “nice” orthogonal matrices

“NICE” lists:

Perfect Triangles

Perfect Triangles

Have you ever noticed that textbooks often involve so-called 3-4-5 triangles? They do that to make the algebraic manipulations easier for students.  If the two legs of a right triangle are of length 3 and 4, then the hypotenuse (found from the Pythagorean theorem) has a length of 5, which is “nice” in the sense that it is an integer rather than an irrational square root that more typically comes from solving the Pythagorean theorem. As discussed in many elementary math sites (such as, another example of a “nice” triangle is the 5-12-13 triangle, since 5^2+12^2=13^2 .

The external links in this posting contain a list of more of these so-called perfect triples of integers \{a,b,c\} for which a^2+b^2=c^2 . Perfect triples are also used to create “nice” 2D unit vectors whose components are each rational numbers (instead of involving irrational square roots from the normalization process). For example, the classic unit vector based on the 3-4-5 perfect triple is simply \{\frac{3}{5},\frac{4}{5}\} .

A related task is to create “nice” 3D unit vectors. Even more interesting is the need to create “nice” 3×3 orthogonal matrices whose components are all rational numbers. Professors of vector analysis and continuum mechanics typically keep a list of “nice” orthogonal matrices to use on exams. Each column of such a matrix is a 3D unit vector, and each column is perpendicular to all of the other columns.
See below for these “nice” lists (of particular value to professors trying to construct homework problems involving “nice” numbers):

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