Question bank — Euler angles — roll φ, pitch θ, yaw ψ; rotation sequence (3-2-1 convention)

False. The name counts the axes down: 3 (yaw about $z$) is applied first, then 2 (pitch about $y$), then 1 (roll about $x$). Roll is applied last.

False. Rotations about different axes do not commute: $R_x{R}_y{R}_z\ne R_z{R}_y{R}_x$. Physically, yawing-then-pitching lands the nose somewhere different than pitching-then-yawing.

True in the sense that they describe the same rotation, but they map opposite directions: $R_n^b$ takes nav→body, while $(R_n^b{)}^T=R_b^n$ takes body→nav. Using the wrong one flips the rotation sense.

False. Orthogonality (columns orthonormal) imposes 6 constraints, leaving exactly 3 degrees of freedom — which is why 3 Euler angles suffice.

False. At $\theta =\pm 90^{\circ }$ (gimbal lock) infinitely many $(\varphi ,\psi )$ pairs give the same orientation — only the sum or difference $\varphi \pm \psi$ is determined, so the map is not one-to-one there.

False. They are transposes of each other. Passive (rotating the frame) puts $-\sin$ in the lower-left; active (rotating the vector) puts $-\sin$ in the upper-right. Same angle, opposite off-diagonal sign.

False. Rotation is nonlinear in the angles — $R(2\varphi ,2\theta ,2\psi )$ is generally not $R(\varphi ,\theta ,\psi {)}^2$ because the three spins are about different, sequentially-defined axes.

Wrong direction. $R_n^b$ maps nav→body. To send a body vector into nav you need $R_b^n=(R_n^b{)}^T$. Applying $R_n^b$ here gives the reverse rotation.

Wrong. $\arcsin$ only returns values in $[-90^{\circ },+90^{\circ }]$. That is actually correct for pitch (physically pitch lives in that range), but you cannot use $\arcsin$ to get roll or yaw — those need $atan2$ to cover all four quadrants.

Wrong. $R_{13}=-\sin \theta$ but $R_{33}=\cos \varphi \cos \theta$, so that ratio is contaminated by $\varphi$. Pitch comes cleanly from $R_{13}$ alone: $\theta =-\arcsin (R_{13})$.

Wrong when the signs matter. Plain $\arctan$ collapses quadrants (it repeats every $180^{\circ }$), so it cannot distinguish $\varphi$ from $\varphi +180^{\circ }$. You must use $atan2(R_{23},R_{33})$, which reads the two signs separately.

Wrong. Gimbal lock is a structural singularity of the Euler parametrization: at $\theta =\pm 90^{\circ }$ two axes physically coincide and a DOF genuinely disappears. No precision helps; you must switch representations (quaternions/DCM).

Wrong order in the product. Applied-first goes on the right so it hits the vector first: $R_n^b=R_x(\varphi )R_y(\theta )R_z(\psi )$. The intrinsic 3-2-1 sequence collapses to the extrinsic product with $R_z$ rightmost.

Because three angles are minimal, human-readable, and match pilot intuition (heading, nose-up, bank). The matrix is redundant (9 numbers, 6 constraints); the angles are the compact 3-DOF core.

Because it mirrors how a vehicle is naturally described: first point the nose (yaw/heading), then raise it (pitch), then bank the wings (roll). Each later spin is about the axis the earlier spins already established.

Because rotating the frame by $+\alpha$ transforms a fixed vector's coordinates by the transpose of the active (vector) rotation. Transposing swaps the off-diagonal sines, moving the negative sign below the diagonal.

Both numerator and denominator of each ratio share it: $R_{23}/R_{33}=(s\varphi c\theta )/(c\varphi c\theta )=\tan \varphi$, and likewise for yaw. The shared $\cos \theta$ divides out — unless $\cos \theta =0$, which is exactly gimbal lock.

Quaternions parametrize rotation on a smooth 4-number sphere with no coordinate singularities, so no orientation ever loses a DOF. Euler angles are three coordinate charts that inevitably fold at $\theta =\pm 90^{\circ }$.

For agile vehicles (fighters, rockets, drones doing loops) pointing straight up is a real, required attitude, not an edge to dodge. You must represent it faithfully, which forces a singularity-free scheme like quaternions or the DCM.

The identity: body frame perfectly aligned with the nav frame ($R_n^b=I$). Nose North, right wing East, belly Down.

The roll axis ($x_b$) and the yaw axis ($z_n$) align, so a change in $\varphi$ and a change in $\psi$ produce the same physical motion. Only their combination is observable — one DOF is lost (gimbal lock).

Yes — angles are periodic with period $360^{\circ }$, so they give identical rotation matrices. This periodicity (not gimbal lock) is a harmless, expected non-uniqueness.

They can represent the same physical attitude by different angle paths — a known non-uniqueness of Euler angles even away from gimbal lock. This is why extracting angles from a DCM needs a fixed convention (e.g. $\theta \in [-90^{\circ },90^{\circ }]$).

It is the body Down axis expressed in nav coordinates; a $z$-component less than 1 means the vehicle is tilted off level. As a row of an orthogonal matrix it is a unit vector, so its magnitude is always exactly 1.

Halfway between North and East in the horizontal plane. With no pitch/roll the nose stays level, and a $+45^{\circ }$ yaw swings it toward the East by half a right angle.