quaternion norm QuaternionNorm 2026-08-23 19:16:02 2026-08-26 03:00:30 Definition updated Related Associations quaternion norm magnitude tensor unit quaternion % this is the default PhysicsLibrary preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \section*{Quaternion Norm} The quaternion norm measures the magnitude of a quaternion in the same way that the Euclidean norm measures the length of a real vector. For \[ q=q_w+q_x\mathbf i+q_y\mathbf j+q_z\mathbf k = q_w+\mathbf q, \] the PhysicsLibrary quaternion norm is \begin{equation} \lVert q\rVert = \sqrt{ q_w^2+q_x^2+q_y^2+q_z^2 }. \end{equation} Using the quaternion conjugate, \[ q^*=q_w-\mathbf q, \] the same quantity can be written \begin{equation} \lVert q\rVert = \sqrt{q q^*} = \sqrt{q^* q}. \end{equation} The squared norm is therefore \begin{equation} \lVert q\rVert^2 = q q^* = q^* q = q_w^2+q_x^2+q_y^2+q_z^2. \end{equation} The norm is always a nonnegative real number. It is zero only for the zero quaternion. The norm is fundamental because it is multiplicative, \begin{equation} \lVert pq\rVert = \lVert p\rVert\,\lVert q\rVert, \end{equation} and because a quaternion with norm one is a \emph{unit quaternion}. Unit quaternions are the objects used in later PhysicsLibrary articles to represent passive frame orientation. \section{Definition} Let \[ q=q_w+\mathbf q \] with \[ \mathbf q=q_x\mathbf i+q_y\mathbf j+q_z\mathbf k. \] The quaternion norm is defined by \begin{equation} \lVert q\rVert = \sqrt{ q_w^2+\mathbf q\cdot\mathbf q }. \end{equation} Since \[ \mathbf q\cdot\mathbf q = q_x^2+q_y^2+q_z^2, \] equation (5) is the ordinary Euclidean norm of the four real quaternion components. In scalar first component notation, \[ [q]_{\mathrm{PL}} = \begin{bmatrix} q_w\\ q_x\\ q_y\\ q_z \end{bmatrix}, \] so \[ \lVert q\rVert \] is exactly the Euclidean length of this four component coefficient vector. This observation concerns the magnitude of the coefficient array. It does not imply that quaternion multiplication is ordinary four dimensional vector multiplication. \section{Derivation from the conjugate} The conjugate of \[ q=q_w+\mathbf q \] is \[ q^*=q_w-\mathbf q. \] Using the Hamilton scalar vector product, \[ (q_w,\mathbf q)(q_w,-\mathbf q) \] has scalar part \[ q_w^2-\mathbf q\cdot(-\mathbf q) = q_w^2+\mathbf q\cdot\mathbf q \] and vector part \[ q_w(-\mathbf q) + q_w\mathbf q + \mathbf q\times(-\mathbf q) = \mathbf 0. \] Therefore \begin{equation} q q^* = q_w^2+\mathbf q\cdot\mathbf q. \end{equation} The same calculation gives \begin{equation} q^*q = q_w^2+\mathbf q\cdot\mathbf q. \end{equation} Hence \[ q q^*=q^*q \] even though quaternion multiplication is not generally commutative. Taking the nonnegative square root gives equation (2). \section{Basic properties} The quaternion norm has the standard properties expected of a norm. \subsection*{Nonnegativity} For every quaternion $q$, \begin{equation} \lVert q\rVert\ge 0. \end{equation} This follows because the squared norm is a sum of squares. \subsection*{Definiteness} The norm vanishes only for the zero quaternion: \begin{equation} \lVert q\rVert=0 \quad\Longleftrightarrow\quad q=0. \end{equation} Indeed, \[ q_w^2+q_x^2+q_y^2+q_z^2=0 \] with real components requires every component to be zero. \subsection*{Real scalar scaling} For a real scalar $a$, \begin{equation} \lVert aq\rVert = |a|\,\lVert q\rVert. \end{equation} \subsection*{Conjugation preserves norm} Since conjugation changes only the signs of the vector components, \begin{equation} \lVert q^*\rVert = \lVert q\rVert. \end{equation} \subsection*{Sign reversal preserves norm} Similarly, \begin{equation} \lVert -q\rVert = \lVert q\rVert. \end{equation} This property later helps explain why $q$ and $-q$ can represent the same physical orientation when $q$ is a unit quaternion. \section{Multiplicative property} One of the most important facts about the quaternion norm is \[ \lVert pq\rVert = \lVert p\rVert\lVert q\rVert. \] A clean proof uses conjugation. Start with \[ \lVert pq\rVert^2 = (pq)(pq)^*. \] Quaternion conjugation reverses product order: \[ (pq)^*=q^*p^*. \] Therefore \[ \begin{aligned} \lVert pq\rVert^2 &= (pq)(q^*p^*)\\ &= p(qq^*)p^*. \end{aligned} \] The quantity $qq^*$ is real, so it commutes with every quaternion: \[ p(qq^*)p^* = (qq^*)pp^*. \] Thus \begin{equation} \lVert pq\rVert^2 = \lVert q\rVert^2 \lVert p\rVert^2. \end{equation} Both norms are nonnegative, so taking square roots gives \begin{equation} \lVert pq\rVert = \lVert p\rVert \lVert q\rVert. \end{equation} The order on the right is irrelevant because the norms are ordinary real numbers. \section{Why multiplicativity matters} The multiplicative norm has several immediate consequences. First, if $p$ and $q$ are both unit quaternions, \[ \lVert p\rVert=\lVert q\rVert=1, \] then \[ \lVert pq\rVert=1. \] Therefore unit quaternions are closed under quaternion multiplication. Second, if $q\neq0$, then \[ \lVert q\rVert>0, \] so the expression \[ \frac{q^*}{\lVert q\rVert^2} \] is well defined. The next PhysicsLibrary article shows that this is precisely $q^{-1}$. Third, norm multiplicativity is a strong implementation check. A software routine intended to compute Hamilton products should satisfy \[ \lVert pq\rVert \approx \lVert p\rVert\lVert q\rVert \] up to numerical roundoff. \section{Unit quaternions} A quaternion is called a \emph{unit quaternion} when \begin{equation} \lVert q\rVert=1. \end{equation} Equivalently, \begin{equation} q_w^2+q_x^2+q_y^2+q_z^2=1. \end{equation} Thus unit quaternions lie on the unit sphere in four dimensional real coefficient space. For a unit quaternion, \begin{equation} q q^*=q^*q=1. \end{equation} Therefore \begin{equation} q^{-1}=q^*. \end{equation} This relation is the key reason unit quaternions are so convenient for orientation calculations. \section{Normalization} Any nonzero quaternion can be converted into a unit quaternion by dividing by its norm: \begin{equation} \widehat q = \frac{q}{\lVert q\rVert}. \end{equation} Then \[ \begin{aligned} \lVert\widehat q\rVert &= \left\lVert \frac{q}{\lVert q\rVert} \right\rVert\\ &= \frac{\lVert q\rVert}{\lVert q\rVert}\\ &=1. \end{aligned} \] Normalization changes the quaternion magnitude while preserving its direction in four dimensional coefficient space. When a quaternion is intended to represent orientation, normalization is often used to remove small numerical drift from floating point propagation. Normalization should not be used blindly to hide a serious integration or convention error. Large norm drift is a diagnostic that the underlying algorithm should be checked. \section{Norm and passive frame orientation} The definition of quaternion norm is algebraic and does not depend on active or passive rotation interpretation. PhysicsLibrary uses a unit quaternion \[ {}^{B}q_A \] to map coordinate components from frame $A$ into frame $B$: \begin{equation} {}^{B}v = {}^{B}q_A\, {}^{A}v\, ({}^{B}q_A)^*. \end{equation} The frame quaternion must satisfy \begin{equation} \lVert{}^{B}q_A\rVert=1. \end{equation} Because of unit norm, \[ ({}^{B}q_A)^{-1} = ({}^{B}q_A)^*. \] Thus the inverse frame map is represented by the conjugate, \[ {}^{A}q_B = ({}^{B}q_A)^*. \] The unit norm constraint is therefore what turns the simple conjugate into the exact inverse needed for a reversible frame coordinate transformation. \section{Norm preservation under passive coordinate mapping} Let $v$ be represented as a pure quaternion. Under the passive coordinate map, \[ {}^{B}v = {}^{B}q_A\,{}^{A}v\,({}^{B}q_A)^*. \] Using norm multiplicativity, \[ \begin{aligned} \lVert{}^{B}v\rVert &= \lVert{}^{B}q_A\rVert \lVert{}^{A}v\rVert \lVert({}^{B}q_A)^*\rVert. \end{aligned} \] For a unit frame quaternion, \[ \lVert{}^{B}q_A\rVert = \lVert({}^{B}q_A)^*\rVert = 1. \] Therefore \begin{equation} \lVert{}^{B}v\rVert = \lVert{}^{A}v\rVert. \end{equation} This is exactly what a change of orthonormal coordinates should do: the physical vector is unchanged, so its Euclidean magnitude is unchanged. \section{Norm and the axis angle form} For a positive frame rotation through angle $\theta$ about unit axis $\widehat{\mathbf u}$, the PhysicsLibrary passive quaternion is \begin{equation} {}^{B}q_A = \cos\frac{\theta}{2} - \widehat{\mathbf u}\sin\frac{\theta}{2}. \end{equation} Its squared norm is \[ \begin{aligned} \lVert{}^{B}q_A\rVert^2 &= \cos^2\frac{\theta}{2} + \lVert\widehat{\mathbf u}\rVert^2 \sin^2\frac{\theta}{2}. \end{aligned} \] Since \[ \lVert\widehat{\mathbf u}\rVert=1, \] we obtain \begin{equation} \lVert{}^{B}q_A\rVert^2 = \cos^2\frac{\theta}{2} + \sin^2\frac{\theta}{2} = 1. \end{equation} Thus the axis angle construction automatically produces a unit quaternion. \section{Historical terminology: tensor and norm} Historical quaternion texts do not always use the modern words \emph{norm} and \emph{squared norm} in the same way as current engineering literature. In much classical quaternion notation, the quantity now written \[ \lVert q\rVert \] was called the \emph{tensor} of the quaternion and written $Tq$. Some historical sources then used the word \emph{norm} for \[ (Tq)^2 = q q^* = q_w^2+q_x^2+q_y^2+q_z^2. \] Modern engineering and applied mathematics usually call \[ \sqrt{ q_w^2+q_x^2+q_y^2+q_z^2 } \] the quaternion norm and call its square the squared norm. PhysicsLibrary follows this modern convention. When reading Hamilton, Tait, Joly, or other historical sources, one should check whether the author's ``norm'' means the modern norm or the modern squared norm. \section{Worked example 1: computing a norm} Let \[ q=2-3\mathbf i+4\mathbf j+5\mathbf k. \] Then \[ \begin{aligned} \lVert q\rVert^2 &= 2^2+(-3)^2+4^2+5^2\\ &= 4+9+16+25\\ &= 54. \end{aligned} \] Therefore \begin{equation} \lVert q\rVert = \sqrt{54} = 3\sqrt6. \end{equation} The conjugate has the same norm: \[ q^* = 2+3\mathbf i-4\mathbf j-5\mathbf k, \] and \[ \lVert q^*\rVert=3\sqrt6. \] \section{Worked example 2: normalization} Let \[ q=1+\mathbf i+\mathbf j+\mathbf k. \] Its norm is \[ \lVert q\rVert = \sqrt{1+1+1+1} = 2. \] Therefore the normalized quaternion is \begin{equation} \widehat q = \frac12 \left( 1+\mathbf i+\mathbf j+\mathbf k \right). \end{equation} A direct check gives \[ \lVert\widehat q\rVert^2 = 4\left(\frac12\right)^2 = 1. \] \section{Worked example 3: multiplicative norm} Take \[ p=1+\mathbf i, \qquad q=1+\mathbf j. \] Their norms are \[ \lVert p\rVert=\sqrt2, \qquad \lVert q\rVert=\sqrt2. \] Their product is \[ pq = 1+\mathbf i+\mathbf j+\mathbf k. \] Hence \[ \lVert pq\rVert = \sqrt{1+1+1+1} = 2. \] The multiplicative property predicts \[ \lVert p\rVert\lVert q\rVert = \sqrt2\sqrt2 = 2, \] which agrees. \section{Worked example 4: passive frame quaternion} Consider a positive $90^\circ$ frame rotation about $+z$. The passive PhysicsLibrary quaternion is \[ {}^{B}q_A = \frac{\sqrt2}{2} - \frac{\sqrt2}{2}\mathbf k. \] Its norm is \[ \begin{aligned} \lVert{}^{B}q_A\rVert^2 &= \frac12+\frac12\\ &=1. \end{aligned} \] Therefore its conjugate is also its inverse: \[ {}^{A}q_B = ({}^{B}q_A)^* = \frac{\sqrt2}{2} + \frac{\sqrt2}{2}\mathbf k. \] \section{Numerical considerations} In exact mathematics, a properly constructed orientation quaternion has norm one. In numerical simulation, finite precision integration may slowly move the stored quaternion away from unit norm. A common numerical correction is \begin{equation} q \leftarrow \frac{q}{\lVert q\rVert}. \end{equation} For small numerical drift this keeps the state on the unit quaternion constraint. Several practical cautions are important: \begin{enumerate} \item Check for a norm near zero before dividing. \item Do not use normalization to compensate for a wrong propagation equation or a frame convention error. \item If norm drift is unexpectedly large, inspect time step size, integration method, angular rate units, multiplication order, and frame definitions. \item When comparing two orientation quaternions, remember that $q$ and $-q$ have the same norm and later will be shown to represent the same orientation. \end{enumerate} \section{Common pitfalls} \begin{enumerate} \item \textbf{Confusing norm with squared norm.} PhysicsLibrary uses \[ \lVert q\rVert = \sqrt{q q^*}, \] not $q q^*$ itself. \item \textbf{Importing historical terminology without checking definitions.} Some classical sources use ``norm'' for what PhysicsLibrary calls the squared norm. \item \textbf{Assuming every quaternion used in algebra has unit norm.} Only unit quaternions satisfy $\lVert q\rVert=1$. \item \textbf{Assuming conjugation changes the norm.} Conjugation preserves norm exactly. \item \textbf{Forgetting norm multiplicativity.} Quaternion multiplication is not commutative, but the norm still satisfies \[ \lVert pq\rVert = \lVert p\rVert\lVert q\rVert. \] \item \textbf{Normalizing the zero quaternion.} The expression $q/\lVert q\rVert$ is undefined for $q=0$. \item \textbf{Changing the norm formula because an attitude convention is passive.} The norm is algebraic and is unchanged by active or passive interpretation. \item \textbf{Treating normalization as a substitute for correct dynamics.} Small numerical drift may be corrected by normalization; large drift should be investigated. \end{enumerate} \section{Exercises} The exercises are stated first so the article can be used for self study. Complete solutions follow afterward. \begin{enumerate} \item \textbf{Direct norm.} For \[ q=3-2\mathbf i+4\mathbf j-\mathbf k, \] compute $\lVert q\rVert^2$ and $\lVert q\rVert$. \item \textbf{Conjugate invariance.} For the quaternion in Exercise 1, compute $q^*$ and verify \[ \lVert q^*\rVert=\lVert q\rVert. \] \item \textbf{Product with conjugate.} Show directly that \[ q q^*=q^*q=\lVert q\rVert^2 \] for \[ q=1+2\mathbf i-\mathbf j+2\mathbf k. \] \item \textbf{Normalization.} Normalize \[ q=2+2\mathbf i+\mathbf j-\mathbf k. \] \item \textbf{Multiplicative norm.} Let \[ p=1+\mathbf i, \qquad q=2+\mathbf j. \] Compute $pq$ and verify \[ \lVert pq\rVert = \lVert p\rVert\lVert q\rVert. \] \item \textbf{Unit quaternion product.} Prove that the product of two unit quaternions is a unit quaternion. \item \textbf{Inverse of a unit quaternion.} Use \[ q q^*=\lVert q\rVert^2 \] to prove that $q^{-1}=q^*$ when $\lVert q\rVert=1$. \item \textbf{Passive frame norm.} Show that \[ {}^{B}q_A = \cos\frac{\theta}{2} - \widehat{\mathbf u}\sin\frac{\theta}{2} \] has unit norm when $\widehat{\mathbf u}$ is a unit vector. \item \textbf{Vector magnitude preservation.} Using multiplicativity of the quaternion norm, prove that the passive map \[ {}^{B}v = {}^{B}q_A\,{}^{A}v\,({}^{B}q_A)^* \] preserves vector magnitude when ${}^{B}q_A$ is unit. \item \textbf{Historical terminology.} A nineteenth century source states that the ``norm'' of \[ q=1+2\mathbf i+2\mathbf j \] is $9$. Is this necessarily inconsistent with the PhysicsLibrary definition? Explain. \end{enumerate} \section{Solutions} \subsection*{1. Direct norm} For \[ q=3-2\mathbf i+4\mathbf j-\mathbf k, \] the squared norm is \[ \lVert q\rVert^2 = 3^2+(-2)^2+4^2+(-1)^2 = 30. \] Therefore \[ \lVert q\rVert=\sqrt{30}. \] \subsection*{2. Conjugate invariance} The conjugate is \[ q^* = 3+2\mathbf i-4\mathbf j+\mathbf k. \] Its squared norm is \[ 3^2+2^2+(-4)^2+1^2 = 30. \] Hence \[ \lVert q^*\rVert = \sqrt{30} = \lVert q\rVert. \] \subsection*{3. Product with conjugate} For \[ q=1+2\mathbf i-\mathbf j+2\mathbf k, \] the squared norm is \[ 1^2+2^2+(-1)^2+2^2 = 10. \] The conjugate is \[ q^* = 1-2\mathbf i+\mathbf j-2\mathbf k. \] By the conjugate product identity, \[ q q^*=q^*q=10. \] \subsection*{4. Normalization} For \[ q=2+2\mathbf i+\mathbf j-\mathbf k, \] the squared norm is \[ 4+4+1+1=10. \] Therefore \[ \widehat q = \frac{1}{\sqrt{10}} \left( 2+2\mathbf i+\mathbf j-\mathbf k \right). \] \subsection*{5. Multiplicative norm} First, \[ \begin{aligned} pq &=(1+\mathbf i)(2+\mathbf j)\\ &=2+2\mathbf i+\mathbf j+\mathbf k. \end{aligned} \] Therefore \[ \lVert pq\rVert^2 = 2^2+2^2+1^2+1^2 = 10, \] so \[ \lVert pq\rVert=\sqrt{10}. \] Also, \[ \lVert p\rVert=\sqrt2 \] and \[ \lVert q\rVert=\sqrt5. \] Hence \[ \lVert p\rVert\lVert q\rVert = \sqrt2\sqrt5 = \sqrt{10} = \lVert pq\rVert. \] \subsection*{6. Unit quaternion product} If \[ \lVert p\rVert=1 \] and \[ \lVert q\rVert=1, \] then multiplicativity gives \[ \lVert pq\rVert = \lVert p\rVert\lVert q\rVert = 1. \] Thus $pq$ is unit. \subsection*{7. Inverse of a unit quaternion} For a unit quaternion, \[ \lVert q\rVert^2=1. \] Therefore \[ q q^*=q^*q=1. \] By definition of multiplicative inverse, \[ q^{-1}=q^*. \] \subsection*{8. Passive frame norm} For \[ {}^{B}q_A = \cos\frac{\theta}{2} - \widehat{\mathbf u}\sin\frac{\theta}{2}, \] the squared norm is \[ \cos^2\frac{\theta}{2} + \lVert\widehat{\mathbf u}\rVert^2 \sin^2\frac{\theta}{2}. \] Since \[ \lVert\widehat{\mathbf u}\rVert=1, \] this becomes \[ \cos^2\frac{\theta}{2} + \sin^2\frac{\theta}{2} = 1. \] Therefore \[ \lVert{}^{B}q_A\rVert=1. \] \subsection*{9. Vector magnitude preservation} Using multiplicativity, \[ \begin{aligned} \lVert{}^{B}v\rVert &= \lVert{}^{B}q_A\rVert \lVert{}^{A}v\rVert \lVert({}^{B}q_A)^*\rVert. \end{aligned} \] A unit quaternion and its conjugate both have norm one. Therefore \[ \lVert{}^{B}v\rVert = \lVert{}^{A}v\rVert. \] \subsection*{10. Historical terminology} For \[ q=1+2\mathbf i+2\mathbf j, \] PhysicsLibrary gives \[ \lVert q\rVert = \sqrt{1+4+4} = 3. \] Its squared norm is \[ \lVert q\rVert^2=9. \] A historical source may call this squared quantity the ``norm'' and call the modern norm the ``tensor.'' Therefore the value $9$ is not necessarily an error; the terminology must be checked. \section{What comes next} The norm and conjugate together give the inverse of every nonzero quaternion: \[ q^{-1} = \frac{q^*}{\lVert q\rVert^2}. \] The next PhysicsLibrary quaternion article derives this formula, explains why left and right division must be treated carefully in a noncommutative algebra, and specializes the result to unit and pure quaternions. Later orientation articles use unit norm as the defining constraint that makes quaternion frame maps invertible by simple conjugation and makes quaternion coordinate transformations preserve physical vector length. \section{Sources and historical notes} Classical quaternion authors often distinguished the \emph{tensor} from the \emph{norm}. In Tait's terminology, the tensor corresponds to the modern quaternion norm, while the historical norm is its square. Modern engineering literature normally uses ``norm'' for the square root quantity. The PhysicsLibrary convention follows modern Euclidean norm terminology and uses \[ \lVert q\rVert = \sqrt{q q^*}. \] The historical distinction is retained here because it is important when reading older quaternion literature. \begin{thebibliography}{9} \bibitem{Hamilton1899} W. R. Hamilton, \emph{Elements of Quaternions}, 2nd ed., edited by C. J. Joly, Longmans, Green, and Co., 1899. Public domain historical source. \PMlinkexternal{Internet Archive scan} {https://archive.org/details/elementsofquater01hamiuoft} \bibitem{Tait1890} P. G. Tait, \emph{An Elementary Treatise on Quaternions}, 3rd ed., Cambridge University Press, 1890. Public domain historical source. \PMlinkexternal{Internet Archive search} {https://archive.org/search?query=An+Elementary+Treatise+on+Quaternions+Tait} \bibitem{Joly1905} C. J. Joly, \emph{A Manual of Quaternions}, Macmillan and Co., London, 1905. Public domain historical source. \PMlinkexternal{Internet Archive scan} {https://archive.org/details/manualofquaterni00jolyrich} \bibitem{Hathaway1896} A. S. Hathaway, \emph{A Primer of Quaternions}, 1896. Public domain historical source. \PMlinkexternal{Project Gutenberg edition} {https://www.gutenberg.org/ebooks/9934} \end{thebibliography} \section*{License} Unless otherwise noted, this PhysicsLibrary entry is intended for release under the Creative Commons Attribution ShareAlike 4.0 International license.