User:Lluisros
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On the linear dependence of points
Oriol's solution provides a more visual proof of the assertion. Here's an alternative proof that extends, perhaps more easily, to points in 3-space. We first give the assertion for points in the plane, and later see how it extends to points in 3-space.
Assertion: Let 
, 
, and 
be three points of the Euclidean plane. Let 
, 
, and 
be three vectors, providing homogeneous coordinates for 
(that is, 
. Then, are aligned if, and only if, 
are linearly dependent.
Proof: Note that are aligned if and only if 
, and 
are linearly dependent. These two vectors are linearly dependent if and only if
But:
Thus, the last determinant also vanishes, which proves the assertion.
Note that the proof easily extends to points in 3-space. Just replace the determinants above, by all minors of the corresponding matrix.
On the power generated by a wrench against a twisting body
