Vector projection in polar cordinatesIn Euclidean Space and Cartesian coordinates system, we know the vector projection of Vector u onto a vector v is simply hat P =((hat u .hat v ) hat v )/(hat v. hat v) What would be the vector projection on polar or spherical coordinates or alternate coordinate systems Suppose we have vectors defined as hat u =u alpha e alpha hat v =v beta e beta Does a projection vector of u on v becomes vecP= (u^alpha v^beta/e_(alpha beta)) \vecv (v^alpha v^ betae_(alpha beta)) Where e alpha represent the basis vector and e_{alpha beta} the metric tensor.

Skye Vazquez

Skye Vazquez

Answered question

2022-09-05

Vector projection in polar cordinates
In Euclidean Space and Cartesian coordinates system, we know the vector projection of Vector u onto a vector v is simply
P = ( u . v )   v v . v
What would be the vector projection on polar or spherical coordinates or alternate coordinate systems?
Suppose we have vectors defined as
u = u α e α
v = v β e β
Does a projection vector of u on v becomes
P = ( u α v β e α β ) v v α v β e α β
Where e α represent the basis vector and e α β the metric tensor.

Answer & Explanation

William Collins

William Collins

Beginner2022-09-06Added 12 answers

That looks correct. The dot product formula for projection doesn't depend on the coordinate system, though you need to know how the dot product actually works, which can be determined by its action on a set of basis vectors.
u v = ( α u α e α ) ( β v β e β )
= α , β u α v β ( e α e β )
= α , β u α v β ( e α β )
But the vectors u and v need to be in the same tangent space (based at the same point) in general. And the vectors are not points like ( r , θ ) in polar coordinates.

Do you have a similar question?

Recalculate according to your conditions!

New Questions in Linear algebra

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?