Question appears to be simple, as it did to me. I'm a bit weak in math but that doesn't help me from improving. During my college physics, I encountered this question and found its solution but can't understand it. Can anyone elaborate the highlighted sections in the pics below?
Three chains act on the bracket such that they create a resultant force having magnitude $F_R$. If two of the chains are subjected to known forces, as shown, determine the orientation $\theta$ of the third chain, measured clockwise from the positive $x$-axis, so that the magnitude of force $F$ in this chain is a minimum. All forces lie in the $x-y$ plane. What is the magnitude of $F$?
Hint: First find the resultant of the two known forces. Force F acts in this direction.
Because the author decided to measure positive angles in the clockwise direction from the positive $x$-axis, The measure of angle $\phi$ is negative. Therefore, since $\boldsymbol{F_1}$ points along the negative $y$-axis, the angle between vectors $\boldsymbol{F_1}$ and $\boldsymbol{F_2}$ is $90^\circ - \phi$.
To minimize the force $\boldsymbol{F}$ required on the third chain, it must point in the same direction as the resultant force $\boldsymbol{F_{R_1}}$, which is why the author says to make $\boldsymbol{F}$ parallel to $\boldsymbol{F_{R_1}}$.