I have read some related posts here and here, but have some confusion after seeing some worked examples in my textbook (Elements of Electromagnetics 7th ed, Sadiku 2018) which seem to disagree with what I understood from those posts.
In one example, there are two vectors $\textbf{A} = 3\textbf{a}_r + 2\textbf{a}_\theta - 6\textbf{a}_\phi$ and $\textbf{B} = 4\textbf{a}_r + 3\textbf{a}_\phi$ denoted in the spherical coordinate system.
By my working, their cross product is $\textbf{A} \times \textbf{B} = 6\textbf{a}_r - 33\textbf{a}_\theta - 8\textbf{a}_\phi$
My confusion lies in finding the magnitude of this vector quantity. Intuitively it seems to me that the magnitude should be $|\textbf{A} \times \textbf{B}| = 6$, the value of the coefficient of $\textbf{a}_r$. Instead, the textbook lists the answer as $|\textbf{A} \times \textbf{B}| = \sqrt{6^2 + (-33)^2 + (-8)^2} = 34.48$.
What is the correct method of finding the magnitude of $|\textbf{A} \times \textbf{B}|$?
Also, is it correct that $|\textbf{A}| = 3$? Or should it be $|\textbf{A}| = \sqrt{3^2 + 2^2 + (-6)^2}=7$?
Thanks,