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The aircraft pitch axis is the transverse body axis about which the nose rotates upward or downward. In the convention used by NASA Glenn Research Center, it passes through the centre of mass and points toward the right wing tip; the aircraft roll axis points toward the nose and the aircraft yaw axis toward the underside. These are body-fixed directions, so their orientation relative to the ground changes with the aircraft.^[ref]

Choose right-handed body coordinates $x$ forward, $y$ right and $z$ down. A force $\mathbf{F}$ acting at displacement $\mathbf{r}$ from the centre of mass produces torque $\mathbf{\tau }=\mathbf{r}\times \mathbf{F}$. Its pitching moment isFor an original tail-load example, $\mathbf{r}=(-4,0,0)\mathrm{m}$ and $\mathbf{F}=(0,0,100)\mathrm{N}$ give ${\tau }_y=+400\mathrm{N}\mathrm{m}$. The downward tail load lifts the nose in this convention. If the aircraft pitch axis is a principal axis, the motion is initially pure pitch and the corresponding moment of inertia is $I_y=2000\mathrm{kg}{\mathrm{m}}^2$, then the instantaneous angular acceleration is $\dot{q}={\tau }_y/I_y=0.20rad/{\mathrm{s}}^2$. Coupled three-axis rotation requires the full Euler's equations.

The axis is a geometrical reference, not necessarily a physical hinge. Changing a control-surface load changes both forces and torques; a zero net translational force does not establish zero pitching moment. A balance measurement therefore needs the reference point and sign convention alongside its reported moment.^[ref]

The aircraft yaw axis is the body-fixed axis about which the nose rotates from side to side. In a right-handed body coordinate system with $x$ forward and $y$ toward the right wing, it is the $z$ axis directed downward through the centre of mass. It completes the orthogonal set with the aircraft roll axis and aircraft pitch axis. The rudder can generate a side force aft of the centre of mass, producing a yawing torque.^[ref]

For a force $\mathbf{F}$ acting at displacement $\mathbf{r}$ from the reference point, the yaw component of $\mathbf{\tau }=\mathbf{r}\times \mathbf{F}$ isIn an original example, a rightward $80\mathrm{N}$ tail force at $\mathbf{r}=(-3,0,0)\mathrm{m}$ gives ${\tau }_z=-240\mathrm{N}\mathrm{m}$. With positive yaw defined nose-right, this force turns the nose left. If the axis is a principal axis with moment of inertia $I_z=1200\mathrm{kg}{\mathrm{m}}^2$ and the initial motion is pure yaw, the instantaneous angular acceleration is $\dot{r}={\tau }_z/I_z=-0.20rad/{\mathrm{s}}^2$. A tail's displacement matters as well as the force magnitude.

The aircraft yaw rate $r$ is a component of body angular velocity; it is not always the rate of change of aircraft heading. A banked vehicle projects its rotation onto several body axes, and its ground track also depends on wind. General coupled motion uses Euler's equations, rather than three independent scalar torque laws. Measuring a heading change alone therefore cannot establish the applied yawing moment.