If force F⃗ = 3î + 4ĵ - 2k̂ acts on a particle position vector 2î + ĵ + 2k̂ then, the torque about the origin will be :

Solution

Physics sir · 4 steps

Torque is r⃗ × F⃗, with r⃗ first. Keep that order in the determinant.

1. Set up. Torque about the origin is τ⃗ = r⃗ × F⃗, with r⃗ = 2î + ĵ + 2k̂ and F⃗ = 3î + 4ĵ - 2k̂. Put r⃗ in the second row and F⃗ in the third.

   τ⃗ = |î, ĵ, k̂; 2, 1, 2; 3, 4, -2|

2. Expand. î: (1)(-2) - (2)(4) = -10. ĵ: -[(2)(-2) - (2)(3)] = -(-10) = 10. k̂: (2)(4) - (1)(3) = 5.
3. Quick check. Torque must be perpendicular to both vectors. τ⃗ · r⃗ = -20 + 10 + 10 = 0 and τ⃗ · F⃗ = -30 + 40 - 10 = 0, so the result is consistent.
4. So. τ⃗ = -10î + 10ĵ + 5k̂, which is option (b).

The idea to keep. Torque is always r⃗ × F⃗, never F⃗ × r⃗. After the cross product, dot the answer with both vectors: both results must be zero.

Answer from the NTA final answer key. Solution by PrepWiser.