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Question: The force \(7\widehat{i} + 3\widehat{j} - 5\widehat{k}\) acts on a particle whose position vector is...

The force 7i^+3j^−5k^7\widehat{i} + 3\widehat{j} - 5\widehat{k} acts on a particle whose position vector is i^−j^+k^\widehat{i} - \widehat{j} + \widehat{k} What is the torque of a given force about the origin?

A

12i^+12j^+10k^12\widehat{i} + 12\widehat{j} + 10\widehat{k}

B

2i^+10j^+12k^2\widehat{i} + 10\widehat{j} + 12\widehat{k}

C

2i^+10j^+10k^2\widehat{i} + 10\widehat{j} + 10\widehat{k}

D

10i^+2j^+k^10\widehat{i} + 2\widehat{j} + \widehat{k}

Answer

12i^+12j^+10k^12\widehat{i} + 12\widehat{j} + 10\widehat{k}

Explanation

Solution

Here, r→=i^−j^+k^\overset{\rightarrow}{r} = \widehat{i} - \widehat{j} + \widehat{k}

F→=7i^+3j^−5k^\overset{\rightarrow}{F} = 7\widehat{i} + 3\widehat{j} - 5\widehat{k}

Torque, τ→=r→×F→\overset{\rightarrow}{\tau} = \overset{\rightarrow}{r} \times \overset{\rightarrow}{F}

τ→=∣i^j^k^1−1173−5∣=i^(5−3)+j^(7−(−5))+k^(3−(−7))\overset{\rightarrow}{\tau} = \left| \begin{matrix} \widehat{i} & \widehat{j} & \widehat{k} \\ 1 & - 1 & 1 \\ 7 & 3 & - 5 \end{matrix} \right| = \widehat{i}(5 - 3) + \widehat{j}(7 - ( - 5)) + \widehat{k}(3 - ( - 7)) or τ→=2i^+12j^+10k^\overset{\rightarrow}{\tau} = 2\widehat{i} + 12\widehat{j} + 10\widehat{k}