By A. S. Smogorzhevski
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Additional info for Acerca de la Geometría de Lobachevski
That is, a triangle with one angle at 90 degrees. 4 Pythagoreantheorem. The variable c is the hypotenuse, which is the longest side of the triangle. This is also the length of the 2D vector (a, b). Given the Pythagorean theorem, the value of c is derived as follows. C2 = a 2 + b 2 Taking the square root of both sides results in c m ~a 2 + b 2 The double vertical bar, II, is used to denote the length of the vector. II c II = "~a2 + b2 Note that this differs from the single vertical bar, I, which is used to denote the absolute value operation (Ic] is the absolute vector).
V Often the length of a vector needs to be compared against some predefined length. 1 ) The squared length can be used to implement this more efficiently. The sqrt function can be quite long to calculate compared to the multiplications and additions required to calculate the dot product. 01 = 0 . 1 2 ) . 01 ) PERPENDICULAR PROJECTION The dot product can be used to project one vector onto another. 6 shows the result of projecting vector a onto vector b. 6 Perpendicularprojection. To calculate the length of vector c, the same trigonometry used to convert from polar coordinates to Cartesian coordinates is used.
10 CrossProduct 39 Another interesting property of the cross product is that it can be used to calculate the sine of the angle between the two vectors a and b. IIa x b II = IIa II IIb IIs i n ( O ) I[~ x bll = sin(O) II, II IIb II The angle between the two vectors, 0, is then calculated as follows. sin l/ ,,axb,,/ IIa II IIb II =0 A common misconception with the cross product is that if both vectors a and b have unit length that the perpendicular vector c will have unit length. This is only true when the angle 0 is ~r/2 (90 degrees).
Acerca de la Geometría de Lobachevski by A. S. Smogorzhevski