The coordinates of a particle with respect to origin in a given reference frame is (1, 1, 1) meters. If a force of F⃗ = î - ĵ + k̂ acts on the particle, then the magnitude of torque (with respect to origin) in z-direction is __________.

Solution

Physics sir · 4 steps

Torque is r⃗ × F⃗, and for the z-part you only need the x and y components.

1. Write the vectors. The position vector is r⃗ = î + ĵ + k̂ and the force is F⃗ = î - ĵ + k̂. Torque about the origin is τ⃗ = r⃗ × F⃗.

   τ⃗ = |î, ĵ, k̂; 1, 1, 1; 1, -1, 1|

2. Pick the z-part. The k̂ component is τ_z = xF_y - yFₓ = (1)(-1) - (1)(1) = -2. The minus sign only says the z-torque points along -k̂.
3. Full vector check. Expanding the whole determinant gives τ⃗ = 2î + 0ĵ - 2k̂, which confirms τ_z = -2.
4. So. The magnitude of the torque in the z-direction is |τ_z| = 2 (in N m when F⃗ is in newtons).

The common slip: 2.83. Reporting the magnitude of the full torque vector, |2î - 2k̂| = 2√2, gives 2.83 instead of only the z-component.

The idea to keep. For one component of r⃗ × F⃗, use the other two axes: τ_z = xF_y - yFₓ. No need to expand the whole determinant.

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