Question

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

• A. \(11\^{i}+19\^{j}−4\^{k}\)
• B. \(-11\^{i}+9\^{j}−16\^{k}\)
• C. \(-17\^{i}+19\^{j}−4\^{k}\)
• D. \(17\^{i}+9\^{j}+16\^{k}\)

Answer 1

The Correct Option is C

To find the torque (\boldsymbol{\tau}) of a force vector about the origin, we use the formula for torque, which is the cross product of the position vector and the force vector:

\[ \boldsymbol{\tau} = \mathbf{r} \times \mathbf{F} \]

Given:

• Force vector, \(\mathbf{F} = 5\mathbf{i} + 3\mathbf{j} - 7\mathbf{k}\)
• Position vector, \(\mathbf{r} = 2\mathbf{i} + 2\mathbf{j} + \mathbf{k}\)

The cross product \(\mathbf{r} \times \mathbf{F}\) can be calculated using the determinant method:

\[ \mathbf{r} \times \mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 2 & 2 & 1 \\ 5 & 3 & -7 \end{vmatrix} \]

Expanding this determinant, we have:

• For the \(\mathbf{i}\) component: \(2(-7) - 1(3) = -14 - 3 = -17\)
• For the \(\mathbf{j}\) component: -(2(-7) - 1(5)) = -( -14 - 5) = 19\)
• For the \(\mathbf{k}\) component: \(2(3) - 2(5) = 6 - 10 = -4\)

Thus, the torque vector \(\boldsymbol{\tau}\) is:

\[ \boldsymbol{\tau} = -17\mathbf{i} + 19\mathbf{j} - 4\mathbf{k} \]

Hence, the correct answer is:

\(-17\mathbf{i}+19\mathbf{j}-4\mathbf{k}\)