A rotation matrix is a matrix which when multiplied by a vector has the effect of changing the direction of the vector but not its magnitude.

Properties

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$\backslash mathcal\{M\}\backslash in\backslash mathbb\{R\}^\{N\backslash times\; N\}$ is a rotation matrix if and only if $\backslash mathcal\{M\}$ is orthonormal.

$\backslash mathcal\{M\}$ is orthonormal if its column vectors form an orthonormal basis of $\backslash mathbb\{R\}^\{N\}$, that is, the scalar product between any two column vectors is zero (orthogonality) and the scalar product of a column vector with itself is unity (normalization).

The inverse of a rotation matrix is its transpose:

$\backslash mathcal\{M\}\backslash ,\backslash mathcal\{M\}^\{-1\}=\backslash mathcal\{M\}\backslash ,\backslash mathcal\{M\}^\backslash top=\backslash mathcal\{I\}$ where $\backslash mathcal\{I\}$ is the identity matrix.

Two dimensions

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In two dimensions, a rotation can be defined by a single angle, $\backslash theta$. Conventionally, positive angles represent anti-clockwise rotation.

The matrix to rotate a column vector in cartesian coordinates about the origin is:

$$

M(\theta) = \begin{pmatrix} \cos{\theta} & -\sin{\theta} \\ \sin{\theta} & \cos{\theta} \end{pmatrix}

Three dimensions

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In three dimensions, a rotation can be defined by three Euler angles, $(\backslash alpha,\backslash beta,\backslash gamma)$, or by a single angle of rotation, $\backslash theta$, and the direction of a vector, $\backslash hat\{\backslash mathbf\{v\}\}\; =\; (x,y,z)$, about which to rotate.

The matrix to rotate a column vector in cartesian coordinates about the origin is:

$M(\backslash alpha,\backslash beta,\backslash gamma)\; =\; \backslash begin\{bmatrix\}$

\cos \alpha \cos \gamma - \sin \alpha \cos \beta \sin \gamma & - \sin \alpha \cos \gamma - \cos \alpha \cos \beta \sin \gamma & \sin \beta \sin \gamma \\ \cos \alpha \sin \gamma + \sin \alpha \cos \beta \cos \gamma & - \sin \alpha \sin \gamma + \cos \alpha \cos \beta \cos \gamma & - \sin \beta \cos \gamma \\ \sin \alpha \sin \beta & \cos \alpha \sin \beta & \cos \beta

\end{bmatrix}

or:

$M(\backslash hat\{\backslash mathbf\{v\}\},\backslash theta)\; =\; \backslash begin\{bmatrix\}$

\cos \theta + (1 - \cos \theta) x^2 & (1 - \cos \theta) x y + (\sin \theta) z & (1 - \cos \theta) x z - (\sin \theta) y \\ (1 - \cos \theta) y x - (\sin \theta) z & \cos \theta + (1 - \cos \theta) y^2 & (1 - \cos \theta) y z + (\sin \theta) x \\ (1 - \cos \theta) z x + (\sin \theta) y & (1 - \cos \theta) z y - (\sin \theta) x & \cos \theta + (1 - \cos \theta) z^2 \end{bmatrix}

Roll, Pitch and Yaw

Taking the second form of this matrix, and substituting the unit vectors $\backslash mathbf\{i\}$, $\backslash mathbf\{j\}$ and $\backslash mathbf\{k\}$ gives the following matricies for rotation about the cartesian axes:

• Rotation around the x-axis:

$$

\mathcal{R}(\theta_R):= \begin{pmatrix} 1 & 0 & 0 \\ 0 & \cos{\theta_R} & \sin{\theta_R} \\ 0 & - \sin{\theta_R} & \cos{\theta_R} \end{pmatrix} where $\backslash theta\_R$ is the roll angle.

• Rotation around the y-axis:

$$

\mathcal{P}(\theta_P):= \begin{pmatrix} \cos{\theta_P} & 0 & - \sin{\theta_P} \\ 0 & 1 & 0 \\ \sin{\theta_P} & 0 & \cos{\theta_P} \end{pmatrix} where $\backslash theta\_P$ is the pitch angle.

• Rotation about the z-axis:

$$

\mathcal{Y}(\theta_Y):= \begin{pmatrix} \cos{\theta_Y} & \sin{\theta_Y} & 0 \\ - \sin{\theta_Y} & \cos{\theta_Y} & 0 \\ 0 & 0 & 1 \end{pmatrix} where $\backslash theta\_Y$ is the yaw angle.

In flight dynamics, the roll, pitch and yaw angles are usually given the symbols $\backslash gamma$, $\backslash beta$, and $\backslash alpha$, respectively, however here the symbols $\backslash theta\_R$, $\backslash theta\_P$, and $\backslash theta\_Y$ are used to avoid confusion with the Euler angles.

Any 3-dimensional rotation matrix $\backslash mathcal\{M\}\backslash in\backslash mathbb\{R\}^\{3\backslash times\; 3\}$ can be characterised by the three angles $\backslash theta\_R$, $\backslash theta\_P$, and $\backslash theta\_Y$, and

$\backslash mathcal\{M\}$ is rotation matrix in $$

\mathbb{R}^{3\times 3}\,\Leftrightarrow\,\exist\,\theta_R,\theta_P,\theta_Y\in[0\ldots\pi):\,\mathcal{M}=\mathcal{Y}(\theta_Y)\,\mathcal{P}(\theta_P)\,\mathcal{R}(\theta_R)

The set of all rotations about a given axis, together with the operation of composition, form a continuous group. The matrices discussed here then provide a representation of the group.

See also

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• Rotation (mathematics)
• Coordinate rotation
• Isometry
• Yaw-pitch-roll system

External links

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• Rotation matrices at Mathworld

Rotational symmetry

Drehmatrix | Rotatiematrix | Матрица вращения