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Combinatus
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Homework Statement
Determine the matrix for the spatial projection perpendicular to the straight line (x1, x2, x3) = t(1, 2, 3). The vector space is orthonormal.
Homework Equations
The Attempt at a Solution
After a trip to #math on freenode that resulted in discussions of Gram-Schmidt processes and bra-ket-like vector notations used in physics, I don't even know where to begin any longer. I suspect that this is supposedly a very rudimentary problem of geometric intuition, since I hadn't heard of projections until yesterday.
Anyway, if u=(a1,a2,a3) is an arbitrary vector and P is the projection (which is assumed to be linear), then P(u) = a1*P(e1) + a2*P(e2) + a3*P(e3), since e1, e2 and e3 are base vectors. Now, P(e1), P(e2) and P(e3) should be column vectors in the sought matrix.
Unfortunately, I don't know how to determine their coordinates.
In one attempt to determine them, I assigned s = (1,2,3) as a vector parallel to the line t(1,2,3). So, the dot product e1*s = |e1||s|cos(A) = 1*1 + 2*0 + 3*0 = 1, if e1=(1,0,0) and A is the angle between e1 and s. Then, A = arccos(1/sqrt(14)), since |s|=sqrt(14).
However, I'm getting nowhere with this approach. I can't really determine a normal of the line t(1,2,3) either. Bleh. Ideas are welcome.
Edit: Perhaps I can use the shortest distance between the tip of the vector u and the line...
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