A force F = 3j N is applied at point r = 2i m from the origin. What is the moment about the origin?
AM = 6i N·m — the moment points in the direction of the position vector
BM = 6j N·m — the moment points in the direction of the force
CM = 6k N·m — from r × F using the right-hand rule
DM = −6k N·m — the moment is negative because the force is in the y-direction
Using M = r × F with r = 2i and F = 3j: the cross product i × j = k, so M = 2·3·(i × j) = 6k N·m. The moment vector points in the +z direction. This illustrates the critical 3D insight: the moment vector points *perpendicular* to both r and F — not in the direction of either. A force in the y-direction at an x-offset produces a rotational tendency about the z-axis. Options A and B represent the common misconception that M points along r or F.
Question 2 Multiple Choice
An engineer computes M_O = (4i + 2j − 3k) N·m as the moment about point O. She wants the moment about a shaft axis defined by unit vector û = j. What is the result, and what type of quantity is it?
AA vector: (4i + 2j − 3k) N·m — the full moment vector projected onto the axis
BA scalar: 2 N·m — the dot product û · M_O extracts the y-component
CA vector: 2j N·m — the j-component of M_O
DA scalar: −3 N·m — the component perpendicular to the shaft
The moment about a specific axis is the scalar M_a = û · M_O = j · (4i + 2j − 3k) = 0(4) + 1(2) + 0(−3) = 2 N·m. This is a scalar, not a vector — it represents the rotational tendency about the chosen axis. The x- and z-components (4i and −3k) are reactions absorbed by the bearing and don't drive rotation about the y-axis. The transition from moment-about-a-point (vector) to moment-about-an-axis (scalar) via dot product is a key operation in 3D statics.
Question 3 True / False
In the cross product formula M_O = r × F, swapping the order to F × r gives a result that is equal in magnitude but opposite in direction.
TTrue
FFalse
Answer: True
The cross product is anti-commutative: F × r = −(r × F). This means swapping operand order flips all three components of the moment vector — same magnitude, opposite direction. In practice, this is one of the most common sign errors in 3D statics. The correct form is r × F (position vector crossed into force), following the right-hand rule: curl fingers from r toward F, thumb points in the direction of M_O.
Question 4 True / False
The moment vector M_O = r × F points in the direction of the applied force F.
TTrue
FFalse
Answer: False
The moment vector is perpendicular to both r and F, by definition of the cross product. It points along the axis about which the force tends to cause rotation — which is generally in a completely different direction from the force itself. For example, a vertical force (in the z-direction) applied at a horizontal offset (in the x-direction) produces a moment vector in the y-direction. The moment vector's direction tells you the rotational axis, not the force direction. Confusing these is the central conceptual error students make when first encountering 3D moments.
Question 5 Short Answer
What is the conceptual difference between the moment of a force about a point and the moment of a force about an axis, and when would you use each?
Think about your answer, then reveal below.
Model answer: The moment about a *point* M_O = r × F is a vector describing the full rotational tendency — both which axis the rotation acts about and how strong it is. The moment about a specific *axis* is the scalar M_a = û · M_O, which isolates just the rotational component acting about one particular direction. You use moment about a point when writing full 3D equilibrium equations (summing all moment components to zero). You use moment about an axis when analyzing a specific shaft, hinge, or pin — to find what torque acts through that constrained direction.
For example, analyzing a bolted plate in 3D requires taking moments about the point of application to write three vector equations. But computing the torque on a specific drive shaft means projecting the moment vector onto the shaft's axis — the other components are reacted by bearings. The axis moment gives the scalar answer a mechanical designer needs: how much torque must the shaft handle?