1. **Problem statement:** You want to move an object relative to its current orientation defined by yaw, pitch, and roll angles. The movement is 6.78125 blocks to the right, 25.5 blocks up (local up), and 14.525 blocks forward. 2. **Understanding the problem:** The local directions (right, up, forward) depend on the object's orientation. To find the global displacement, you need to transform the local vector $\vec{v} = \begin{bmatrix}6.78125 \\ 25.5 \\ 14.525\end{bmatrix}$ by the rotation matrix derived from yaw, pitch, and roll. 3. **Rotation matrices:** - Yaw ($\psi$) rotates around the global up axis (usually Y-axis): $$R_{yaw} = \begin{bmatrix}\cos\psi & 0 & \sin\psi \\ 0 & 1 & 0 \\ -\sin\psi & 0 & \cos\psi\end{bmatrix}$$ - Pitch ($\theta$) rotates around the local right axis (X-axis): $$R_{pitch} = \begin{bmatrix}1 & 0 & 0 \\ 0 & \cos\theta & -\sin\theta \\ 0 & \sin\theta & \cos\theta\end{bmatrix}$$ - Roll ($\phi$) rotates around the local forward axis (Z-axis): $$R_{roll} = \begin{bmatrix}\cos\phi & -\sin\phi & 0 \\ \sin\phi & \cos\phi & 0 \\ 0 & 0 & 1\end{bmatrix}$$ 4. **Combined rotation matrix:** The total rotation matrix $R$ is the product of these three matrices. The order depends on convention, but a common order is yaw, then pitch, then roll: $$R = R_{yaw} \times R_{pitch} \times R_{roll}$$ 5. **Calculate global displacement:** Multiply the local displacement vector by $R$: $$\vec{d}_{global} = R \times \vec{v}$$ 6. **Summary:** - Convert yaw, pitch, roll angles to radians. - Compute $R_{yaw}$, $R_{pitch}$, $R_{roll}$. - Multiply to get $R$. - Multiply $R$ by $\vec{v}$ to get the global displacement vector. This gives the position offset in global coordinates based on the object's orientation and local movement vector.