Finding Polar & Azimuthal Angles of 3D Vector

[susantha]

Hi,
Can anybody help me to find the polar angle and the azimuthal angle of a given 3D vector? Say we know the ordinates of the two point A(x1,y1,z1) and B(x2,y2,z2). So the question is find angles (Theta and Phi) of the vector AB. Any suggestion would be really appreciated.

Thanks.
Susantha

 

[tiny-tim]

welcome to pf!

hi susantha! welcome to pf!

(have a theta: θ and a phi: φ and try using the X₂ icon just above the Reply box )

drop a perpendicular from (x,y,z) to the x,y plane … that gives you tanθ, then φ wil be the usual planar polar coordinate

 

[HallsofIvy]

You can, of course, calculate the vector from A to B just by subtracting coordinates. You can then calculate the unit vector by dividing each component by the length of the vector.

And the components of that unit vector are the "direction cosines". That is, if the unit vector is [itex]a\vec{i}+ b\vec{j}+ c\vec{k}[/itex] and the angle the vector makes with the x-axis is [itex]\theta[/itex], the angle the vector makes with the y-axis is [itex]\phi[/itex], and the angle the vector makes with the z-axis is [itex]\psi[/itex], then

[tex]a= cos(\theta)[/tex],
[tex]b= cos(\phi)[/tex],
and
[tex]c= cos(\psi)[/tex]

Of course, those angles are not independent: [itex]cos^2(\theta)+ cos^2(\phi)+ cos^2(\psi)= 1[/itex] so it is sufficient to know two.

If I understand you correctly, the "polar angle" is [itex]\theta[/itex] and the "azimuthal angle" is [itex]\psi[/itex].

 

[susantha]

Thank you for your valuable comments. Now I guess I have the algorithm to find two angles (teta and phi) of a given 3d vector. Here is the algorithm:

// First find the vector AB. (vector AB= vector B - vector A). B=(x1,y1,z1) and A=(x2,y2,z2)
// vector AB =(del_x, del_y, del_z);
del_x = x1-x2;
del_y = y1-y2;
del_z = z1-z2;
// Then we can find the magnitude of the vector AB,
r=sqrt (del_x^2 + del_y^2 +del_z^2);
s = sqrt ( del_x^2 + del_y^2 );
teta = arccos(del_z / r); // teta is the polar angle

//to find the azimuth angle,
if (del_x > 0)
phi= arcsin( del_y / s);
else
phi=pi - arcsin( del_y / s);

if (del_x > 0 && del_y <0 )
phi= pi - phi;

The last if condition is to make sure the phi has positive values between zero and 2*pi.

Thanks again for all comments.
Susantha

 

[PJC01]

Hello Susantha,

To find the polar angle and azimuthal angle of a 3D vector, you can use the following formulas:

Polar angle (Theta) = arccos(z/r)

Azimuthal angle (Phi) = arctan(y/x)

Where r is the magnitude of the vector (AB), which can be calculated using the Pythagorean theorem:
r = √((x2-x1)^2 + (y2-y1)^2 + (z2-z1)^2)

I hope this helps. Let me know if you have any further questions or need clarification. Good luck with your calculations!

Best,