Final answer:
The magnitude of the torque τd caused by a force vector about axis d is calculated using the formula T = rF sin θ, taking into account the distance r from the pivot point to the force application and the perpendicular lever arm r1.
Step-by-step explanation:
The torque τd about axis d due to a force f can be calculated using the relationship T = rF sin θ where T is the torque, r is the distance from the pivot point to the point where the force is applied, F is the magnitude of the force, and θ is the angle between the force and the vector directed from the point of application to the pivot point. In your case, the axis d is at a distance d from the origin and makes an angle φ with the x-axis. The perpendicular lever arm r1 is defined as r1 = r sin θ. To compute the torque, it's essential that the angle θ used in the torque equation is the angle between the force vector and the lever arm, which might differ from the angle φ given in the problem description.