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Find the torque \(\tau = \mathbf{r} \times \mathbf{F}\) of a force \(\mathbf{F} = 3\hat{i} - \hat{j} + 5\hat{k}\) acting at the point \(\mathbf{r} = 7\hat{i} + 3\hat{j} + \hat{k}\).

1. \(14\hat{i} - 38\hat{j} + 16\hat{k}\)
2. \(4\hat{i} + 4\hat{j} + 6\hat{k}\)
3. \(14\hat{i} - 38\hat{j} - 16\hat{k}\)
4. \(21\hat{i} - 3\hat{j} - 5\hat{k}\)

Correct answer: \(14\hat{i} - 38\hat{j} + 16\hat{k}\)

Solution

Torque is given by the vector cross product \(\boldsymbol{\tau}=\mathbf{r}\times\mathbf{F}\). Substituting \(\mathbf{r}=(7,3,1)\) and \(\mathbf{F}=(3,-1,5)\), the cross product gives \((14,-38,16)\).

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