I have a physics problem that states
|| What is the magnetic field at the position of the dot in Figure ex $32.5$ ? Give your answer as a vector.
The correct solution is (according to the book) using The Biot-Savart law,
My question is why is the book choosing to use $\sin(135^\circ)$ ? I understand this has something to do with perspective and convention. My assumption is convention would say the angle of theta starts from the positive $x$ axis. I want to understand why this solution uses the angle of theta from what appears the negative $x$ axis. As $\sin(135^\circ)$ is noted from WolframAlpha below.
Because the field produced by a charge q moving in direction v at position r follows the law
B = muo/4pi * q/|r|^2 * v x r
where B, v, and r are all vector quantities and x is the cross-product. The magnitude of a cross-product AxB is equal to the magnitudes of A and B multiplied by the angle between vectors A and B.
So in the diagram, you show, you want to measure the angle theta relative to the direction of the velocity vector (which in this case is upwards). If you rotated the whole diagram by any arbitrary angle, you should still use the same theta value - it is independent of the x and y axes you use.