For a dynamics class, I have observed a weird correlation which my Professor couldn't explain. So I was wondering if someone would give me an explanation of what's happening .
Here's a general example to explain my query.
Given two Vectors $$ \hat{\mathbf{a}}=3\mathbf{i}+\mathbf{k} ,\hat{\mathbf{b}} =2\mathbf{i}+2\mathbf{k} $$ at planes situated at $$ \mathbf{j}=2,\mathbf{j}=4 $$ , we are required to find the magnitude and angle $\theta $ of a vector $\mathbf{F}$ situated in plane at $$ \mathbf{j}=6 $$ such that $$ (2\mathbf{j} \times \hat{\mathbf{a}}) +4\mathbf{j} \times (\hat{\mathbf{b}}) = 6\mathbf{j}\times \mathbf{F}$$
Weirdly enough(to me) I found that the difference between the angle calculated via the polar form of the L.H.S of the above equation and 90 degrees gives me the correct angle $\theta$ required.
Question Is there a valid reason as to why such a thing even works?
It's simpler than you think. Look at this diagram:
You found that $\theta_F = \theta_{ab} - 90$. But we know that:
$$ (2\mathbf{j} \times \hat{\mathbf{a}}) +(4\mathbf{j} \times \hat{\mathbf{b}}) = 6\mathbf{j}\times \mathbf{F} $$
so $\theta_{ab} = \theta_{6j\times F}$. Because $\vec{6j}$ is normal to the $ik$ plane the vector $6j \times \vec{F}$ lies in the $ik$ plane at 90 degrees to $\vec{F}$. That automatically means that to get from $\theta_{6j\times F}$ to $\theta_F$ you rotate by -90 degrees.