The torque of a force $5 \hat{i}+3 \hat{j}-7 \hat{k}$ about the origin is $\tau$. If the force acts on a…

The torque of a force $5 \hat{i}+3 \hat{j}-7 \hat{k}$ about the origin is $\tau$. If the force acts on a particle whose position vector is $2 \hat{i}+2 \hat{j}+\hat{k}$, then the value of $\tau$ will be :
  1. $11 \hat{\mathrm{i}}+19 \hat{\mathrm{j}}-4 \hat{\mathrm{k}}$
  2. $-11 \hat{\mathbf{i}}+9 \hat{\mathbf{j}}-16 \hat{\mathrm{k}}$
  3. $-17 \hat{\mathrm{i}}+19 \hat{\mathrm{j}}-4 \hat{\mathrm{k}}$
  4. $17 \hat{i}+9 \hat{j}+16 \hat{k}$

Solution

Given that torque, $\vec{\tau}=5 \hat{i}+3 \hat{j}-7 \hat{k}$
Position vector, $\overrightarrow{\mathrm{r}}=2 \hat{\mathrm{i}}++2 \hat{\mathrm{j}}+\hat{\mathrm{k}}$
$\begin{aligned} & \vec{\tau}=\overrightarrow{\mathrm{r}} \times \overrightarrow{\mathrm{F}}=\left|\begin{array}{ccc}\mathrm{i} & \mathrm{j} & \mathrm{k} \\ 2 & 2 & 1 \\ 5 & 3 & -7\end{array}\right| \\ & =\hat{\mathrm{i}}(-14-3)-\hat{\mathrm{j}}(-14-5)+\hat{\mathrm{k}}(6-10) \\ & =-17 \hat{\mathrm{i}}+19 \hat{\mathrm{j}}-4 \hat{\mathrm{k}}\end{aligned}$

Asked in: BITSAT 2023 (Memory Based Paper 2)

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