How to Find the Resultant Force from Multiple 3D Vectors?

[Skittlz]

Homework Statement

The vertices of a quadrilateral are A(1,2,-1) B(-3,2,-3) C(4,1,-4) and D(2,-1,3). At A forces of magnitude 4, 5 and 4 are Newtons act along AB, AC, and AD respectively. Find the resultant force.

Looking for more of a hint than an answer.

Any help would be appreciated x

 

[lucasem_]

The problem essentially gave you the direction of three forces, and their magnitudes. From these you could write the three forces as vectors, \vec{F_1}, and then add the vectors to give you the direction and magnitude of the resultant net force

 

[tiny-tim]

Welcome to PF!

Hi Skittlz! Welcome to PF!

Start by writing out the three vectors …

show us what you get. ​

 

[Skittlz]

By adding I get AB = (-2,4,-4) AC = (5,3,-5) and AD = (3,1,2) would I then add these 3 vectors to get the direction and then use Pythagoras to find the magnitude?

Edit: this is wrong.

 

[tiny-tim]

Hi Skittlz!

By adding I get AB = (-2,4,-4) AC = (5,3,-5) and AD = (3,1,2)

No, for the vector joining A and B, you need to subtract, (not add) …

the vector AB is OB - OA (where O is the origin), ie the coordinates of B minus the coordinates of A.

… would I then add these 3 vectors to get the direction and then use Pythagoras to find the magnitude?

Yes.

(but you will stil need to use moments, to find the correct line of application, or at least any point on that line)

 

[Skittlz]

I see, thanks!

One thing I still don't understand. what am I supposed to do with the given forces?

 

[tiny-tim]

… what am I supposed to do with the given forces?

eg the force along AB is of magnitude 4 …

the vector AB (found by subtraction) tells you the direction, and you then have to multiply that by something (it isn't 4 ! ) to get the actual force …

show us what you get ​

 

[Skittlz]

I know not to multiply by 4 as that just makes the vector larger i.e 4 times larger in this case. But 4N is the force across the vector which would mean |AB| = 4 right? So given that AB = (4,0,2) - (by subtracting) the magnitude would be srt20 right? but we are told that it is 4 so do we multiply by arbitrary values of x,y,z so that srt(4x^2+0y^2+2z^2) = 4 ? or is this wrong?

I feel like I'm missing something quite important here but not sure what - vectors are a very big problem for me :(

 

[tiny-tim]

HiSkittlz!

Think in terms of unit vectors.

So given that AB = (4,0,2) - (by subtracting) the magnitude would be srt20 right?

Right

So the unit vector along AB is (4/√20, 0 2/√20) …

that's what you multiply by 4 ! ​

 

[Skittlz]

Ah I see now, so what i do is:

Find each unit vector then multiply by the force across it e.g.

unit vector of AB = 4 x (4/√20, 0, 2/√20)

Then do the same for the other two and then add the forces to find the resultant force.

 

[tiny-tim]

yup!

(and use moments to find the line of application)

 

[Skittlz]

Thanks for the help :D