The torque of a force $5 \hat{i} + 3 \hat{j} - 7 \hat{k}$ about the origin is $τ$ . If the force acts on a particle whose position vector is $2 \hat{i} + 2 \hat{j} + \hat{k},$ then the value of $τ$ will be

Option 1 - <math> <mrow> <mn>1</mn> <mn>1</mn> <mover accent="true"> <mi>i</mi> <mo>^</mo> </mover> <mo>+</mo> <mn>1</mn> <mn>9</mn> <mover accent="true"> <mi>j</mi> <mo>^</mo> </mover> <mo>−</mo> <mn>4</mn> <mover accent="true"> <mi>k</mi> <mo>^</mo> </mover> </mrow> </math>

Option 2 - <math> <mrow> <mo>−</mo> <mn>1</mn> <mn>1</mn> <mover accent="true"> <mi>i</mi> <mo>^</mo> </mover> <mo>+</mo> <mn>9</mn> <mover accent="true"> <mi>j</mi> <mo>^</mo> </mover> <mo>−</mo> <mn>1</mn> <mn>6</mn> <mover accent="true"> <mi>k</mi> <mo>^</mo> </mover> </mrow> </math>

Option 3 - <math> <mrow> <mo>−</mo> <mn>1</mn> <mn>7</mn> <mover accent="true"> <mi>i</mi> <mo>^</mo> </mover> <mo>+</mo> <mn>1</mn> <mn>9</mn> <mover accent="true"> <mi>j</mi> <mo>^</mo> </mover> <mo>−</mo> <mn>4</mn> <mover accent="true"> <mi>k</mi> <mo>^</mo> </mover> </mrow> </math>

Option 4 - <math> <mrow> <mn>1</mn> <mn>7</mn> <mover accent="true"> <mi>i</mi> <mo>^</mo> </mover> <mo>+</mo> <mn>9</mn> <mover accent="true"> <mi>j</mi> <mo>^</mo> </mover> <mo>+</mo> <mn>1</mn> <mn>6</mn> <mover accent="true"> <mi>k</mi> <mo>^</mo> </mover> </mrow> </math>

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Alok Kumar Singh | Contributor-Level 10

6 months ago

Correct Option - 3

Detailed Solution:

$\vec{F} = 5 \hat{i} + 3 \hat{j} - 7 \hat{k}$

$\vec{r} = 2 \hat{i} + 2 \hat{j} + \hat{k}$

$\vec{τ} = | \vec{r} * \vec{F} | = | \begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ 2 & 2 & 1 \\ 5 & 3 & - 7 \end{matrix} | \begin{matrix} = \hat{i} (- k_{1} - 3) - \hat{j} (- 14 - 5) + \hat{k} (6 - 10) & = - 17 \hat{i} + 19 \hat{j} - 4 \hat{k} \end{matrix}$