Find Force Vector for Given Position & Torque

[DOMINGO79]

I am having a huge problem trying to answer the following question:
Given the position vector r = 3i + 4j + 5k and torque = 16i - 20j - 5k, find the force vector F which will give the correct result, so that torque = r x F
Can anyone please help me solve this!

 

[amcavoy]

I am having a huge problem trying to answer the following question:
Given the position vector r = 3i + 4j + 5k and torque = 16i - 20j - 5k, find the force vector F which will give the correct result, so that torque = r x F
Can anyone please help me solve this!

Write out the cross product as a determinant with the force vector's components a,b,c. Then you can solve:

$$\det{\begin{bmatrix}\mathbf{i}&\mathbf{j}&\mathbf{k} \\ 3&4&5 \\ a&b&c\end{bmatrix}}=16\mathbf{i}-20\mathbf{j}-5\mathbf{k}$$

 

[DOMINGO79]

16i-20j-5k = i j k = i(4c-5b)-j(3c-5a)+k(3b-4a)
3 4 5
a b c

16i = i(4c-5b) -3(16 = 4c-5b) -48 = -12c + 15b
-20j = -j(3c-5a) → 4(20 = 3c-5a) → 80 = 12c – 20a
-5k = k(3b-4a) 32 = 15b -20a

32 = 15b-20a 32 = 15b-20a
5(-5 = 3b-4a) → -25 = -15b+20a, so what am I doing wrom?

why are the unknowns totally canceling out?

 

[Tide]

They don't cancel out. You're not expanding the determinant correctly.

For example, your first equation should read: 16 = 4c - 5b and nothing else.

 

[DOMINGO79]

Wow...

Hey, thanks a lot, I think I just opened my eyes and understand how to complete the question...

If I have anymore Q's, I will let you know, thanks...