In unit-vector notation, what is the torque about the origin on a jar of jalapen

Question

In unit-vector notation, what is the torque about the origin on a jar of jalapeno peppers located at coordinates (3.0 m, -2.0 m, 4.0 m) due to force Upper F1 = (2.1 N) i - (9.7 N) j + (3.6 N) k ((a), (b) and (c) for i , j and k components respectively), force F2 = (-2.1 N) i - (9.7 N) j - (3.6 N) k ((d), (e) and (f) for i , j and k components respectively), and the vector sum of F1 and F2 ((g), (h) and (i) for i , j and k components respectively)? Repeat previous part for the torque about the point with coordinates (3.0 m, 2.0 m, 4.0 m) ((j), (k) and (l) for i , j and k components

Explanation / Answer

torque T = force X perpendicular distance

so

T = F x r

part 1:

T = (2.1 i - 9.7 j + 3.6 k) x (3 i - 2 j +4 k)

T = i j k

2.1 -9.7 3.6

3 -2 4

T = i((-9.7 * 4)+(3.6 * 2)) -j((2.1*4 - 3.6*3)) + k ((2.1*2) -(9.7 *3))

T = -31.6 i + 2.4 j -24.9 k

so   

a = -31.6 i

b = + 2.4 j

c = -24.9 k

------------------------------------------------------------

T2 = F2 x R

F2 = -2.1 i -9.7 j -3.6 k

T2 = i j k   

-2.1 -9.7 -3.6

3 -2 4

T2 = i((-9.7 * 4)-(3.6 * 2)) -j((-2.1*4 + 3.6*3)) + k ((-2.1*2) +(9.7 *3))

T2 = - 46 i + 2.4 j + 24.9 k

so

d = -46 i

e = +2.4 j

f = +24.9 k
-----------------------------------------

g = -31.6 -46 = - 77.6 i

h = 2.4 +2.4 = 4.8 j

i = -24.9 + 24.9 = 0 k

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