Discussion
A vector \(\vec{F} = 5\hat{i} - 10\hat{j} + 8\hat{k}\) is passing through the origin of a 3-D frame. Considering the tendency of rotation in the counter clockwise direction as positive, the moment about a point A: (3, 4, 8) is
Answer: D
The moment is \(-112\hat{i} - 16\hat{j} + 50\hat{k}\).
The moment of a force about a point A is \(\vec{r} \times \vec{F}\), where \(\vec{r}\) runs from A to any point on the force's line of action. The force passes through the origin, so \(\vec{r} = O - A = (-3, -4, -8)\).
\[ \begin{aligned} M_{x} &= (-4)(8) - (-8)(-10) = -112 \\ M_{y} &= (-8)(5) - (-3)(8) = -16 \\ M_{z} &= (-3)(-10) - (-4)(5) = 50 \end{aligned} \]
Using \(\vec{r}\) from the origin to A, (3, 4, 8), reverses every sign and gives \(112\hat{i} + 16\hat{j} - 50\hat{k}\), the moment about the origin of a force through A. The other two options rearrange the same three numbers into the wrong components.
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