# A force F  = 4i – 5j  + 3k  is acting a point (r_1 )  = i  + 2j  + 3k . The torque acting about a point (r_2 )  = 3i  – 2j  – 3k  is

Q: A force $\vec{F} = 4\hat{i} – 5 \hat{j} + 3 \hat{k}$ is acting a point $\vec{r_1} = \hat{i} + 2 \hat{j} + 3 \hat{k}$ . The torque acting about a point $\vec{r_2 } = 3 \hat{i} – 2 \hat{j} – 3 \hat{k}$ is

(a) zero

(b) $42 \hat{i} – 30 \hat{j} + 6 \hat{k}$

(c) $42 \hat{i} + 30 \hat{j} + 6 \hat{k}$

(d) $42 \hat{i} + 30 \hat{j} – 6 \hat{k}$

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Ans: (d)
Sol: $\displaystyle Torque \; \vec{\tau} = \vec{r} \times \vec{F}$

$\displaystyle \vec{r} = \vec{r_1} – \vec{r_2}$

$\displaystyle \vec{r} = (\hat{i} + 2 \hat{j} + 3 \hat{k} ) – (3 \hat{i} – 2 \hat{j} – 3 \hat{k})$

$\displaystyle \vec{r} = (- 2\hat{i} + 4 \hat{j} + 6 \hat{k} )$

$\displaystyle \vec{\tau} = (- 2\hat{i} + 4 \hat{j} + 6 \hat{k} ) \times (4\hat{i} – 5 \hat{j} + 3 \hat{k} )$