Angles in a Triangle and Vectors

Question

Given are two forces $P_1$ and $P_2$ which both act upon a particle point P, and the angle between the two forces is 40 degrees. Furthermore $P_1$ has an absolute value of 20 and $P_2$ of 12. What is the absolute value of the resulting force and what is the angle between the resulting force and $P_1$?

Answer 1

Let $P_1$ be $20\pmatrix{1 \\0}$ and $P_2=12\pmatrix{\cos(40^\circ) \\ \sin(40^\circ)}$ and then do some vector addition to get the resulting force vector $$ P_3=20\pmatrix{1 \\0}+12\pmatrix{\cos(40^\circ) \\ \sin(40^\circ)} $$ and to get the angle between $P_3$ and $P_1$, calculate the inverse cosine of the dot product $$ \arccos\frac{P_3 \cdot P_1}{||P_3|| ||P_1||} $$