I am writing a space searching algorithm but, since I'm just an engineer, my maths obviously isn't great ...
I have a vector, v0, in n-dimensional space and I want to search in, say, 4 directions radially out from a point, p0, on the vector.
So, I know the vector, I know the point and I can quite easily compute the equation of the hyperplane (in an ax+by+cz+d=0 format) that is normal to the vector and on which p0 sits.
Now I want to find the vectors in the plane. Here's what I did, it doesn't seem like a very elegant way and, although it works in a 3D case, I don't see how to make it work in my real case (3 to 10 dimensions).
I found a new point, p1, on plane by arbitrarily fixing x and y and calculating z from equation of plane and then found a vector along plane by v1 = p1-p0. This gives me one of my 4 directions and times by -1 gives me another in the opposite direction.
Now to get the other two directions, I know this vector, v2 is normal to both v0 and v1 so I form two simultaneous equations from the dot product equations. I have 3 unknowns and only two equations so I arbitrarily set one coefficient, a, to 1 and solve the equations. This did give me v2 and hence my other 2 directions.
So, I'm hoping a clever maths person can tell me:
1. Is my method valid (the arbitrary setting of coeffs seems a bit dodgy!)?
2. Would it work for n-dimensional case (would I have to arbitrarily set n-2 coeffs which seems impossible?)?
3. Is there some simple, clever method out there that I could use instead?!
Many thanks in anticipation for any help you can provide.
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