# Determining the $z$-component of a force to maintain equilibrium?

I came across this problem in vector mechanics wherein I’m asked to determine the value of $$d$$ such that the tension in cables $$AC$$ and $$AD$$ is half the tension in $$AB$$.

What I initially did was figure out the forces acting on point $$A$$ by multiplying the magnitude of the tensions by their respective unit vectors to formulate an system of equilibrium equations. But along the way, it became apparent to me that this may be the wrong approach because the unit vector of the tension in $$AB$$ will involve a variable, and frankly, I don’t know how to even start solving that.

The second method I tried was to express the tension in $$AB$$ as $$\left$$ to eliminate the existence of d in the system initially and just solve for the magnitude of the components first and use the relationship of the tension in $$AC$$, $$AD$$, and $$AB$$ (tension in $$AC$$ and $$AD$$ must be equal to half the tension in $$AB$$) to express either $$AC$$ or $$AD$$ as one variable but when I plug it into the system, one equation will simply be zero, thus returning a math error from my calculator.

As a last resort, I tried expressing the tensions $$AC$$ and $$AD$$ in terms of tension $$AB$$ outright, but it gave me a vector with a $$z$$ component only.

I’m really running out of ideas to try. I also don’t fully grasp the unit vector notation in $$AB$$ to denote its orientation so that might be the reason why I can’t answer this problem. If anyone could lead me to the right path, I’d really appreciate it. Thank you!

• Equate the force component in the horizontal (xy) plane. The numbers (6,2,3) have been carefully chosen to make the arithmetic/trigonometry easier to do. – Farcher Aug 28 '19 at 7:59
• Welcome to this community! I just added a few cosmetics to the math statements to make reading easier. – engineer Aug 28 '19 at 8:45

Hint: Try rotating the coordinates so that $$AB$$ is on the x-axis. That eliminates $$AB_y$$ and $$AB_z$$ and $$d$$ from $$AB$$ and adds $$d$$ to the z components of $$AD$$ and $$AC$$. I don’t know if that will help, but that’s where I’d start.