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

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Vector projection in non-orthogonal coordinates

**Physics Forums | Science Articles, Homework Help, Discussion**