Suppose I had a plane and for whatever reason, I chose two non-orthogonal vectors in R3 to define that plane (they define a basis for the plane?). Suppose I have another vector in that plane. How do I find the (contravariant?) coordinates of another arbitrary vector in that plane? All I want to do is decompose the vector along my vectors defining that plane s1 and s2, but simply dotting component-wise by saying(adsbygoogle = window.adsbygoogle || []).push({});

v = dot(s1, v) * s1_vec + dot(s2,v) * s2_vec

Doesn't look right geometrically because it maps the unknown vector V onto the basis vectors in Euclidian sense, instead of a curvilinear sense.

Is the solution to this to use the metric tensor defined by Jtranspose J and use that to replace the Cartesian dot product?

Thanks for any help

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# Vector projection in non-orthogonal coordinates

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