Why a cross product? Concept
A force doesn't just push or pull — if it's applied away from a support or pivot, it also twists that point. That twisting effect is called the moment of the force about that point.
Two things decide how hard it twists:
- How big the force is, and how it's angled
- How far away — and in what direction — it's applied, measured by the position vector r from the support to the point of application
The cross product packages both of these into a single vector operation:
r and F are both vectors. Their cross product gives a third vector — the moment — that's perpendicular to both, with a magnitude and a direction that tell you exactly how the support is being twisted.
r runs from the support O to where F is applied. The moment MO curls around O, out of the page.
Computing it: the determinant method Mechanics
Write r = (rx, ry, rz) and F = (Fx, Fy, Fz). The cross product is the determinant of this matrix:
| rx ry rz |
| Fx Fy Fz |
Expanding it gives three components:
My = rzFx − rxFz
Mz = rxFy − ryFx
For a purely planar problem — r and F both lying in the xy-plane, which is the usual beam/bracket case — the first two components vanish and only Mz survives:
Worked example Walkthrough
A bracket is bolted to a wall at O. A cable pulls with force F = (40, −30, 0) N at a point located at r = (0.5, 0.2, 0) m from the bolt.
Mz = (0.5)(−30) − (0.2)(40)
Mz = −15 − 8 = −23 N·m
The negative sign means the bolt feels a clockwise moment of 23 N·m. That's the number an engineer checks against the bolt's rated torque capacity.
Worked example 2: cantilever beam Walkthrough
A beam is welded into a wall at O and sticks straight out along x. A downward load is hung at its free end, 2 m out from the wall.
F = (0, −500, 0) N
Mz = (2)(−500) − (0)(0)
Mz = −1000 N·m
Negative again: the weld feels a clockwise moment of 1000 N·m. This is exactly the bending moment you'd check against the weld's strength — the further out the load sits, the bigger r, the bigger the twist, even though the force never changed.
Fixed beam, load at the tip. The clockwise curl shows the beam wants to droop and twist the wall joint clockwise.
Worked example 3: pushing a door open Walkthrough
A door hinge sits at O. You push perpendicular to the door, 0.8 m from the hinge, with 60 N, aimed to swing the door open.
F = (60, 0, 0) N
Mz = (0)(0) − (0.8)(60)
Mz = −48 N·m
The hinge takes a 48 N·m clockwise moment — that's the swing torque. Push at the same spot but angle your hand 45° off perpendicular, and only the perpendicular component of F still does the twisting; the part of F aimed straight at the hinge contributes nothing, because it's parallel to r.
Door swinging open about its hinge. r runs up the door to your hand; F is your push, perpendicular to the door for maximum effect.
Build your own r × F Interactive
Enter a position vector r (in metres, from the support) and a force vector F (in newtons). We'll compute the full moment vector and walk through the determinant.
Quick checks Self-test
Five short ones. Three come with a picture of the actual object — work the vectors out from the diagram before you calculate.
r = (0.3, 0, 0) m, F = (0, 50, 0) N. What is Mz?
You pull straight up on a wrench handle, 0.2 m from the bolt, with 100 N applied perpendicular to the handle.
F = (0, 100, 0) N
A beam is fixed into a wall at O and a load hangs straight down at its tip, 1.5 m out.
F = (0, −200, 0) N
Hinge at O. You push perpendicular to the door, 0.9 m from the hinge, with 40 N.
F = (40, 0, 0) N
r = (0.6, −0.2, 0) m, F = (10, 30, 0) N. What is Mz?
What does the magnitude of r × F physically represent?
r = (0.4, 0, 0) m, F = (0, 25, 0) N, both in the xy-plane. Find Mz in N·m.
r = (0.2, 0.6, 0) m, F = (10, −20, 0) N. Find Mz in N·m.
A calculated Mz comes out negative. What does that mean about the support?
A beam fixed at O carries a downward load at its tip, 3 m out.
F = (0, −150, 0) N
Find Mz in N·m.
You pull straight up on a wrench handle, 0.2 m from the bolt, with 100 N perpendicular to the handle (same setup as Practice, Check 2). Which way does the bolt get twisted?
How does F × r compare to r × F?
r = (1, 1, 0) m, F = (5, −5, 0) N. Find Mz in N·m.
Hinge at O. Push perpendicular to the door, 0.9 m out, with 40 N (same setup as Practice, Check 4).
F = (40, 0, 0) N
Find Mz in N·m.
The line of action of F passes directly through the support O. What is the moment about O?
A signpost is bolted to the ground at O. Wind load hits the sign plate at a point offset in all three directions.
F = (10, −20, 15) N
Find Mx, My, and Mz in N·m.
r and F both have x, y, and z components — none of M_x, M_y, M_z drops out this time.
An L-shaped bracket is fixed to a wall at O. It extends out along x then up along z, and carries a load there.
F = (0, 40, −10) N
Find Mx, My, and Mz in N·m.
r bends through x and z; F sits in the y–z plane. Compute the determinant carefully, term by term.