Cartesian Vector Form - Door with 2 Chains

[BuckBee]

Homework Statement

[/B]
The door is held open by the means of 2 chains. If the tension in AB and CD is Fa = 300 N and Fc = 250 N, respectively, express each of these in Cartesian Vector Form

Homework Equations

Sin / cos / tan

The Attempt at a Solution

The angle of FA at B is atan(1.5sin30/(1+1.5cos30 )) then +y direction force A is 300cos that angle and -z direction force is 300sin that angle

There are 2 forces in 2D but I don't understand the 3D. There are 3 forces x, y and z

See Image Attached

 

[haruspex]

There are 2 forces in 2D but I don't understand the 3D.

Take it in stages. Drop a vertical from C to a point G below it on the ground. The tension has a vertical component and a horizontal one from G to D.

 

[BuckBee]

Thanks but I am still a bit confused with the (x) force. I am really not sure about the 3D calculations. The horizontal length for C to D is 2 meters with a Fc of 200N

+y direction force is tan(1.5sin30/(1+1.5cos30) = X then 300 CosX

-z direction force is tan(1.5sin30/(1+1.5cos30) = X then 300 SinX

 

[haruspex]

The horizontal length for C to D is 2 meters

No, that's just the X component. What is the whole horizontal distance? Hence what is tan(∠CDG)?

 

[BuckBee]

ok I admit I have no idea how to do the 3D x, y, z forces for CD. I have no idea what formula to use or where to start

The 2D y, z forces for AB I understand that, see the calculations in the image attached below, I am sure that's correct

 

[BuckBee]

 

[haruspex]

how to do the 3D x, y, z forces for CD.

C, D, G lie in a vertical plane. That's just 2D.
Use Pythagoras to find the distance DG, hence find ∠CDG.

 

[BuckBee]

cool thanks but we don't know the distance DC to find out the length DG using Pythagoras

 

[BuckBee]

ok got it!

 

[BuckBee]

So does that look correct? If so I can then work out the x, y, z forces. Also how do I express the answer in cartesian vector form?

 

[haruspex]

So does that look correct? If so I can then work out the x, y, z forces. Also how do I express the answer in cartesian vector form?

Yes, that's it. You might want to keep one more digit of precision during the calculation... I have 3.14 where you have 3.1.
Cartesian vector form probably means as three numbers in parentheses: (Fx, Fy, Fz).

 

[BuckBee]

Great, thanks for the help, I think I've got it now. Can you please double check the calculations and answers and let me know if correct, Cheers

 

[BuckBee]

 

[BuckBee]

 

[haruspex]

I'm getting rather different numbers. Where does the 2.23 come from? 1+0.75(√3)/2=2.3.
But you've basically cracked it, so I'll show you an easier way. You do not need to calculate the angles in degrees.
In 2D, the components of a unit force in the direction (x,y) are x/h, y/h where h=√(x²+y²), right?
In 3D, that extends simply to x/h, y/h, z/h where h=√(x²+y²+z²).