How can I find Va given Vo and angle A on a 3D sphere?

  • Thread starter Thread starter Ed
  • Start date Start date
  • Tags Tags
    3d Angles
AI Thread Summary
To find the vector Va given the unit vector Vo and angle A on a 3D sphere, one must understand that Va is perpendicular to Vo and lies within the plane tangent to the sphere at point P. The angle A is defined between Va and the vector Vz, which connects point P to the Z-axis. By assuming point P is on the x-axis, the problem can be simplified to basic 2D trigonometry. The transformation to Cartesian coordinates involves calculating the coordinates of Va based on the defined angle and then translating these coordinates to point P. This approach effectively reduces the complexity of the problem while allowing for the determination of Va.
Ed
Messages
12
Reaction score
0
Hi folks, I'm hoping for a little help with something which I'm sure should be pretty easy... but I don't have access to any kind of maths textbooks to look anything up, sadly, so I'm hoping you guys might help?

Let's consider a point P on the surface of a sphere, and define Vo as the unit vector from the origin of the sphere to this point. This vector also describes the normal to the plane which is tangential to the sphere at P.

Now let Va be a (unit) vector within this plane, i.e. Va is perpendicular to Vo, and the angle A describes the angle subtended by Va and the point within the plane which intersects the sphere's +Z axis.

Another way of looking at this would be: if Vz is the vector between P and the Z axis, within the plane normal to Vo, the angle A is between Va and Vz.

So now the question... if Vo and A are known... how do I get Va? (in terms of cartesian co-ords)

edit: made it a little clearer, I hope, and added a diagram drawn in MS Word which I hope should also help.
 

Attachments

Last edited:
Mathematics news on Phys.org
It looked easy using an angle transformation... (You know the polar coors? Good.) Just assume that P is on the x-axis (or y if you want) and Va is now perpendicular to x right? Proceed after this and finally transfer (xva, 0, 0) to your P point. (Later, you can find the carthesian coordinates by tr.(so that wasnt `finally`) Sorry for not being clear about the transformations as i have little time.
 
You're quite right, by assuming it's on an axis it makes it possible to reduce the problem to a couple of 2D basic trig problems. Thanks for the reply.
 
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...
Thread 'Imaginary Pythagoras'
I posted this in the Lame Math thread, but it's got me thinking. Is there any validity to this? Or is it really just a mathematical trick? Naively, I see that i2 + plus 12 does equal zero2. But does this have a meaning? I know one can treat the imaginary number line as just another axis like the reals, but does that mean this does represent a triangle in the complex plane with a hypotenuse of length zero? Ibix offered a rendering of the diagram using what I assume is matrix* notation...
Back
Top