Computing vector to equal vector forces/magnitudes I have this problem with its respective solution:

Assuming that the superior vector is $\overrightarrow{A}$, the middle vector is $\overrightarrow{B}$, and the inferior vector is $\overrightarrow{F}$
My computation was:
$$\overrightarrow{A} \cdot \overrightarrow{B}=\overrightarrow{B} \cdot \overrightarrow{F}\\
|\overrightarrow{A}||\overrightarrow{B}|cos(40)=|\overrightarrow{B}||\overrightarrow{F}|cos(35)\\
|\overrightarrow{F}| = \frac{20000*cos(40)}{cos(35)} \space\space\space\space \text{(given} |\overrightarrow{A}|=20000 \text{)} \\
|\overrightarrow{F}| = 18703
$$
Where am I wrong? I'm looking for a vector solution.
 A: You've used cosine in your calculations where you should have used sine. You are calculating the magnitude of $\vec{F}$ such that the horizontal components of $\vec{F}$ and $\vec{A}$ would be equal. Replacing the $\cos$ in your expressions with $\sin$ provides the right answer.
You can remember by the SOH CAH TOA mnemonic that sine provides the side opposite the angle (in this case the vertical component) and cosine provides the side adjacent (in this case the horizontal component).
A: Although I don't recommend it for this application, one can get a vector solution using $$\vec{A}\times \vec{B}=S\vec{F}\times \vec{B}$$
Where variable $S$ is the unknown force.
$$
\vec{A} \, =  \, \left( \begin{align}20000 \; \cos \left( 2 \cdot \frac{\pi }{9} \right) \\ 20000 \; \sin \left( 2 \cdot \frac{\pi }{9} \right) \\ 0 \end{align} \right)$$
$$
\vec{B} \, =  \, \left( \begin{align}1 \\ 0 \\ 0 \end{align} \right)
$$
$$
\vec{F} \, =  \, \left( \begin{align}S \; \operatorname{cos} \left( 7 \cdot \frac{\pi }{36} \right) \\ S \; \operatorname{sin} \left( 7 \cdot \frac{\pi }{36} \right) \\ 0 \end{align} \right)
$$

Now $$\vec{A}\times\vec{B}=\left( \begin{align}0 \\ 0 \\ -20000 \; \sin \left( \frac{2}{9} \; \pi  \right) \end{align} \right)
$$
$$\vec{F}\times\vec{B}=\left( \begin{align}0 \\ 0 \\ -S \; \sin \left( \frac{7}{36} \; \pi  \right) \end{align} \right)$$
Solving for $S$
$$ \left\{ S = 20000 \cdot \frac{\sin \left( 2 \cdot \frac{\pi }{9} \right)}{\sin \left( 7 \cdot \frac{\pi }{36} \right)} \right\}=22413.32 $$
Just because you can, doesn't mean you should.
