Moments with vectors and Varignon's theorem
In three dimensions, the cross product gives a moment's size and direction at once. Varignon's theorem lets you break forces into convenient pieces.
The cross product
The moment of a force F about point O is \(\mathbf{M}_O = \mathbf{r} \times \mathbf{F}\), where r is a position vector from O to any point on F's line of action. The cross product's magnitude is \(rF\sin\theta\) and its direction is perpendicular to both r and F, following the right-hand rule.
Computing the cross product
In components, \(\mathbf{r} \times \mathbf{F} = (r_yF_z - r_zF_y)\mathbf{i} - (r_xF_z - r_zF_x)\mathbf{j} + (r_xF_y - r_yF_x)\mathbf{k}\), usually evaluated as a 3 by 3 determinant. For r = (2, 0, 0) m and F = (0, 0, 50) N, the moment is (0, -100, 0) N·m: 100 N·m about the negative y-axis.
Order matters
The cross product is not commutative: \(\mathbf{F} \times \mathbf{r} = -(\mathbf{r} \times \mathbf{F})\). Always write r first, from the moment point to the force. Using the wrong order flips the moment's direction, a common sign error.
Any point on the line
The moment does not depend on which point of the line of action r reaches, because sliding the force along its line does not change its turning effect. This principle of transmissibility lets you pick the most convenient point, often where the line crosses an axis.
Varignon's theorem
Varignon's theorem states that the moment of a force about a point equals the sum of the moments of its components about that point: \(\mathbf{r} \times (\mathbf{F}_1 + \mathbf{F}_2) = \mathbf{r} \times \mathbf{F}_1 + \mathbf{r} \times \mathbf{F}_2\). Pierre Varignon published it around 1687. In 2D, it justifies the formula \(M = xF_y - yF_x\).
A 2D example
A 200 N force at 30 degrees above horizontal acts at the point (2, 0) m. Its components are about 173.2 N and 100 N. About the origin, the horizontal component passes through O's horizontal line and has no moment, so M = 2 x 100 = 200 N·m anticlockwise. Finding the perpendicular distance directly would be slower.
Moment about an axis
Often the turning about a specific axis matters, such as a door's hinge line. The moment about an axis with unit vector u is the scalar \(M_a = \mathbf{u} \cdot (\mathbf{r} \times \mathbf{F})\), the component of the moment vector along the axis. A force parallel to an axis, or one whose line crosses it, has zero moment about that axis.
Worked example
A pipe wrench is turned by a force F = (0, 0, -60) N at the handle end, r = (0.25, 0.1, 0) m from the pipe's centre. Then r x F = (0.1 x -60 - 0, -(0.25 x -60 - 0), 0) = (-6, 15, 0) N·m. About the pipe's axis, the y-axis, the twisting moment is 15 N·m.
Key takeaways
The vector moment is M = r x F, with r from the moment point to any point on the force's line; its direction follows the right-hand rule. Compute cross products by determinant and keep the order r then F, since swapping reverses the sign. Transmissibility means any point on the line of action works. Varignon's theorem (around 1687) says a force's moment equals the sum of its components' moments, giving M = x Fy - y Fx in 2D. The moment about an axis is u dot (r x F).
Flashcard
M = r x F
Check your understanding
A 200 N force at 30 degrees above horizontal acts at (2, 0) m. Its moment about the origin is: