Final answer:
The magnitude of the moments of a force around the axes in Engineering can be found using the cross product of the position and force vectors, applying the relation for torque, and considering vector directions and trigonometry.
Step-by-step explanation:
To determine the magnitude of the moments of a force about the x, y, and z axes using the vector approach, one must consider the force vector and its positioning relative to the point of rotation. The moment (or torque) about an axis is given by the cross product of the position vector (from the axis to the point of force application) and the force vector itself. The magnitude of this moment is the product of the force, the perpendicular distance from the axis to the line of action of the force (lever arm), and the sine of the angle between them.
In the provided information, the force vectors F1 and F2 have components along the x, y, and z axes. By applying the relation for torque (||=r₁F), where r₁ is the lever arm, we can find the moments about each axis. This involves calculating the cross product of the position vector and the force vector, then finding the magnitudes of these moments separately for the x, y, and z axes.
It is crucial to use proper vector operations here, including vector addition and subtraction, as well as scalar multiplication when necessary. Other important considerations are the directions of the force components and ensuring that the correct trigonometric functions are used when finding the moments.