In unit-vector notation, what is the net torque about the origin on a flea located at coordinates (-2.0, 4.0 m, -1.0 m) when forces F1 = (-4.0 N) k and F2 = (-5.0 N) j act on the flea?
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Torque is the cross product of the radius vector and the net force vector. In three dimensions, it can be computed as fallows.
|i... j... k|
|0... -4... -5|
|-2... 4... -1|
Compute the determinant
= 20 Nm
The determinant is = (4 * -1 - (-5) * 4)i - (0 * -1 - (-5) * -2)j + (0 * 4 - (-4) * -2)k
= (-4 - (-20))i - (0 - 10)j + (0 - 8)k
= 16i - 10j - 8k
Therefore, the net torque about the origin on the flea is (16 N)m in the x-direction, (-10 N)m in the y-direction, and (-8 N)m in the z-direction.
To find the net torque about the origin, we need to calculate the cross product of the radius vector and the net force vector.
The radius vector points from the origin to the location of the flea, given by (-2.0, 4.0 m, -1.0 m).
The net force vector is the sum of forces F1 and F2, given as F1 = (-4.0 N) k and F2 = (-5.0 N) j, respectively.
To calculate the cross product, we can use the determinant method. Write the three unit vectors (i, j, k) in the first row, the components of the net force vector (-4, 0, -5) in the second row, and the components of the radius vector (-2, 4, -1) in the third row:
|i... j... k|
|0... -4... -5|
|-2... 4... -1|
Now, we need to compute the determinant of this 3x3 matrix:
Det = (0 * (-1) * 4) + (0 * 4 * (-2)) + ((-4) * (-1) * (-2)) - ((-5) * 4 * 0) - ((-5) * (-1) * (-2)) - (0 * (-4) * (-2))
= 0 + 0 + 8 - 0 - 10 - 0
= -2 N.m
The net torque about the origin on the flea is -2 N.m in unit-vector notation.