If force $\hat{\mathrm{F}} = 3\hat{i} + 4\hat{j} - 2\hat{k}$acts on a particle having position vector $2\hat{\mathrm{i}} +\hat{\mathrm{j}}+2\hat{\mathrm{k}}$ then, the torque about the origin will be :-

Given the force F and the position vector r of a particle, the torque τ about the origin can be calculated using the cross product formula:

τ = r × F

Step 1: Identify the Position and Force Vectors

The position vector r is given as:

r = 2i + 1j + 2k

The force vector F is given as:

F = 3i + 4j - 2k

Step 2: Set up the Cross Product

To calculate the torque, we need to compute the cross product r × F. This can be represented as the determinant of a 3x3 matrix:

    τ = |i   j   k|
        |2   1   2|
        |3   4  -2|
    

Step 3: Calculate the Determinant

To expand the determinant, we apply the rule for the cross product:

    τ = i |1   2|  - j |2   2|  + k |2   1|
               |4  -2| |3  -2|       |3   4|
    

Sub-calculations:

• For i:

  |1   2|  = (1)(-2) - (2)(4) = -2 - 8 = -10

• For -j:

  |2   2|  = (2)(-2) - (2)(3) = -4 - 6 = -10, and so, -(-10) = 10

• For k:

  |2   1|  = (2)(4) - (1)(3) = 8 - 3 = 5

Step 4: Combine the Results

Putting it all together, we get:

τ = -10i + 10j + 5k

Final Result

The torque about the origin is:

τ = -10i + 10j + 5k