Make two dependent vectors point in a certain direction when chained in 2D

Question

Ok, I got two vectors $A$ and $B$, and $E=A+B$. I got a vector $T$ pointing in my target direction. I want $E$ to point in the same direction as $T$. Now the tricky part is that the angle of $A$ is $\alpha$ and the angle of $B$ is $\frac 5 3 \alpha$. Also, $|A| \ne |B|$.

Below is an illustration including an equation I got so far, however I'm not able to solve that equation. Known values: $|A|, |B|, T$.

Can anyone help me out? Thanks.

Answer 1

You need to solve this numerically. The equation that needs to be solved is

$$ \tan ( \epsilon - \alpha ) = \frac{B \sin \left( \frac{2}{3} \alpha \right)}{A+B \cos \left( \frac{2}{3} \alpha \right) } $$

The simplest method would be single point iteration. Starting from $\alpha_0=0$ do

$$ \alpha_i = \epsilon - \tan^{-1} \left( \frac{B \sin \left( \frac{2}{3} \alpha _{i-1}\right)}{A+B \cos \left( \frac{2}{3} \alpha_{i-1} \right) } \right)$$

It should converge within 7 to 12 iterations to enough accuracy.

---

Example:

$A=5$, $B=2$, $\epsilon = 60°$

1. $\alpha_0 = 0$
2. $\alpha_1 = 1.04719755119659 {\rm rad}$
3. $\alpha_2 = 0.852871937781031$

$\vdots$

4. $\alpha_{12} = 0.882200775986220 {\rm rad} = 50.5463811471765°$

Appendix

Lay a coordinate system along A and measure along to get $$E \cos(\epsilon-\alpha) = A + B \cos(\frac{2}{3}\alpha) $$

Take the perpendicular direction and measure $$ E \sin(\epsilon-\alpha) = B \sin(\frac{2}{3} \alpha) $$

Divide the two to get the solution.