The choice of the cosine function ensures that when \ langle u, \ v \ rangle = 0 \, ( orthogonal vectors ), the angle ? = ? / 2 or-? / 2, where the sign is determined by an orientation on the vector space.
22.
The vertical reaction force F = mg with which the ramp pushes back to your sled can be decomposed into two orthogonal vectors of, respectively, magnitude F�cos ? ( directed upwards normal to the ramp ) and F�sin ? ( directed backwards along the ramp ), where ? stands for the slope angle of the ramp.
23.
This is due to the two-to-one correspondence between SU ( 2 ) and SO ( 3 ) mentioned above, and the fact that, although one visualizes spin up / down as the north / south pole on the 2-sphere 2, they are actually represented by orthogonal vectors in the two dimensional complex Hilbert space.
24.
The second is zero because it is the dot product of orthogonal vectors, the third is zero because it contains the divergence of flow velocity, and the fourth is zero because the divergence of a vector with only component three is zero ( since it is assumed that nothing ( except maybe h _ 3 ) depends on component three ).
25.
:: : : This restriction corresponds to the OP's " I know that we can define an ellipsoid by its center and a set of orthogonal vectors . " That is to say, you need to be able to freely choose the orthogonal vectors " B " ( up to order and sign ) and the centre.
26.
:: : : This restriction corresponds to the OP's " I know that we can define an ellipsoid by its center and a set of orthogonal vectors . " That is to say, you need to be able to freely choose the orthogonal vectors " B " ( up to order and sign ) and the centre.
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