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Unit vector

From Wikipedia, the free encyclopedia
Vector of length one
Not to be confused withVector of ones.

Inmathematics, aunit vector in anormed vector space is avector (often aspatial vector) oflength 1. A unit vector is often denoted by a lowercase letter with acircumflex, or "hat", as inv^{\displaystyle {\hat {\mathbf {v} }}} (pronounced "v-hat"). The termnormalized vector is sometimes used as a synonym forunit vector.

Thenormalized vector û of a non-zero vectoru is the unit vector in the direction ofu, i.e.,

u^=u‖u‖=(u1‖u‖,u2‖u‖,...,un‖u‖){\displaystyle \mathbf {\hat {u}} ={\frac {\mathbf {u} }{\|\mathbf {u} \|}}=\left({\frac {u_{1}}{\|\mathbf {u} \|}},{\frac {u_{2}}{\|\mathbf {u} \|}},...,{\frac {u_{n}}{\|\mathbf {u} \|}}\right)}

where ‖u‖ is thenorm (or length) ofu andu=(u1,u2,...,un){\textstyle \mathbf {u} =(u_{1},u_{2},...,u_{n})}.[1][2]

The proof is the following:‖u^‖=u1u12+...+un22+...+unu12+...+un22=u12+...+un2u12+...+un2=1=1{\textstyle \|\mathbf {\hat {u}} \|={\sqrt {{\frac {u_{1}}{\sqrt {u_{1}^{2}+...+u_{n}^{2}}}}^{2}+...+{\frac {u_{n}}{\sqrt {u_{1}^{2}+...+u_{n}^{2}}}}^{2}}}={\sqrt {\frac {u_{1}^{2}+...+u_{n}^{2}}{u_{1}^{2}+...+u_{n}^{2}}}}={\sqrt {1}}=1}

A unit vector is often used to representdirections, such asnormal directions.Unit vectors are often chosen to form thebasis of a vector space, and every vector in the space may be written as alinear combination form of unit vectors.

Orthogonal coordinates

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Cartesian coordinates

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Main article:Standard basis

Unit vectors may be used to represent the axes of aCartesian coordinate system. For instance, the standardunit vectors in the direction of thex,y, andz axes of athree dimensional Cartesian coordinate system are

x^=[100],y^=[010],z^=[001]{\displaystyle \mathbf {\hat {x}} ={\begin{bmatrix}1\\0\\0\end{bmatrix}},\,\,\mathbf {\hat {y}} ={\begin{bmatrix}0\\1\\0\end{bmatrix}},\,\,\mathbf {\hat {z}} ={\begin{bmatrix}0\\0\\1\end{bmatrix}}}

They form a set of mutuallyorthogonalunit vectors, typically referred to as astandard basis inlinear algebra.

They are often denoted using commonvector notation (e.g.,x orx→{\displaystyle {\vec {x}}}) rather than standard unit vector notation (e.g.,x̂). In most contexts it can be assumed thatx,y, andz, (orx→,{\displaystyle {\vec {x}},}y→,{\displaystyle {\vec {y}},} andz→{\displaystyle {\vec {z}}}) are versors of a 3-DCartesian coordinate system. The notations (î,ĵ,k̂), (x̂1,x̂2,x̂3), (êx,êy,êz), or (ê1,ê2,ê3), with or withouthat, are also used,[1] particularly in contexts wherei,j,k might lead to confusion with another quantity (for instance withindex symbols such asi,j,k, which are used to identify an element of a set or array or sequence of variables).

When a unit vector in space is expressed inCartesian notation as a linear combination ofx,y,z, its three scalar components can be referred to asdirection cosines. The value of each component is equal to the cosine of the angle formed by the unit vector with the respective basis vector. This is one of the methods used to describe theorientation (angular position) of a straight line, segment of straight line, oriented axis, or segment of oriented axis (vector).

Cylindrical coordinates

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See also:Jacobian matrix

The threeorthogonal unit vectors appropriate to cylindrical symmetry are:

They are related to the Cartesian basisx^{\displaystyle {\hat {x}}},y^{\displaystyle {\hat {y}}},z^{\displaystyle {\hat {z}}} by:

ρ^=cos⁡(φ)x^+sin⁡(φ)y^{\displaystyle {\boldsymbol {\hat {\rho }}}=\cos(\varphi )\mathbf {\hat {x}} +\sin(\varphi )\mathbf {\hat {y}} }
φ^=−sin⁡(φ)x^+cos⁡(φ)y^{\displaystyle {\boldsymbol {\hat {\varphi }}}=-\sin(\varphi )\mathbf {\hat {x}} +\cos(\varphi )\mathbf {\hat {y}} }
z^=z^.{\displaystyle \mathbf {\hat {z}} =\mathbf {\hat {z}} .}

The vectorsρ^{\displaystyle {\boldsymbol {\hat {\rho }}}} andφ^{\displaystyle {\boldsymbol {\hat {\varphi }}}} are functions ofφ,{\displaystyle \varphi ,} and arenot constant in direction. When differentiating or integrating in cylindrical coordinates, these unit vectors themselves must also be operated on. The derivatives with respect toφ{\displaystyle \varphi } are:

∂ρ^∂φ=−sin⁡φx^+cos⁡φy^=φ^{\displaystyle {\frac {\partial {\boldsymbol {\hat {\rho }}}}{\partial \varphi }}=-\sin \varphi \mathbf {\hat {x}} +\cos \varphi \mathbf {\hat {y}} ={\boldsymbol {\hat {\varphi }}}}
∂φ^∂φ=−cos⁡φx^−sin⁡φy^=−ρ^{\displaystyle {\frac {\partial {\boldsymbol {\hat {\varphi }}}}{\partial \varphi }}=-\cos \varphi \mathbf {\hat {x}} -\sin \varphi \mathbf {\hat {y}} =-{\boldsymbol {\hat {\rho }}}}
∂z^∂φ=0.{\displaystyle {\frac {\partial \mathbf {\hat {z}} }{\partial \varphi }}=\mathbf {0} .}

Spherical coordinates

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The unit vectors appropriate to spherical symmetry are:r^{\displaystyle \mathbf {\hat {r}} }, the direction in which the radial distance from the origin increases;φ^{\displaystyle {\boldsymbol {\hat {\varphi }}}}, the direction in which the angle in thex-y plane counterclockwise from the positivex-axis is increasing; andθ^{\displaystyle {\boldsymbol {\hat {\theta }}}}, the direction in which the angle from the positivez axis is increasing. To minimize redundancy of representations, the polar angleθ{\displaystyle \theta } is usually taken to lie between zero and 180 degrees. It is especially important to note the context of any ordered triplet written inspherical coordinates, as the roles ofφ^{\displaystyle {\boldsymbol {\hat {\varphi }}}} andθ^{\displaystyle {\boldsymbol {\hat {\theta }}}} are often reversed. Here, the American "physics" convention[3] is used. This leaves theazimuthal angleφ{\displaystyle \varphi } defined the same as in cylindrical coordinates. TheCartesian relations are:

r^=sin⁡θcos⁡φx^+sin⁡θsin⁡φy^+cos⁡θz^{\displaystyle \mathbf {\hat {r}} =\sin \theta \cos \varphi \mathbf {\hat {x}} +\sin \theta \sin \varphi \mathbf {\hat {y}} +\cos \theta \mathbf {\hat {z}} }
θ^=cos⁡θcos⁡φx^+cos⁡θsin⁡φy^−sin⁡θz^{\displaystyle {\boldsymbol {\hat {\theta }}}=\cos \theta \cos \varphi \mathbf {\hat {x}} +\cos \theta \sin \varphi \mathbf {\hat {y}} -\sin \theta \mathbf {\hat {z}} }
φ^=−sin⁡φx^+cos⁡φy^{\displaystyle {\boldsymbol {\hat {\varphi }}}=-\sin \varphi \mathbf {\hat {x}} +\cos \varphi \mathbf {\hat {y}} }

The spherical unit vectors depend on bothφ{\displaystyle \varphi } andθ{\displaystyle \theta }, and hence there are 5 possible non-zero derivatives. For a more complete description, seeJacobian matrix and determinant. The non-zero derivatives are:

∂r^∂φ=−sin⁡θsin⁡φx^+sin⁡θcos⁡φy^=sin⁡θφ^{\displaystyle {\frac {\partial \mathbf {\hat {r}} }{\partial \varphi }}=-\sin \theta \sin \varphi \mathbf {\hat {x}} +\sin \theta \cos \varphi \mathbf {\hat {y}} =\sin \theta {\boldsymbol {\hat {\varphi }}}}
∂r^∂θ=cos⁡θcos⁡φx^+cos⁡θsin⁡φy^−sin⁡θz^=θ^{\displaystyle {\frac {\partial \mathbf {\hat {r}} }{\partial \theta }}=\cos \theta \cos \varphi \mathbf {\hat {x}} +\cos \theta \sin \varphi \mathbf {\hat {y}} -\sin \theta \mathbf {\hat {z}} ={\boldsymbol {\hat {\theta }}}}
∂θ^∂φ=−cos⁡θsin⁡φx^+cos⁡θcos⁡φy^=cos⁡θφ^{\displaystyle {\frac {\partial {\boldsymbol {\hat {\theta }}}}{\partial \varphi }}=-\cos \theta \sin \varphi \mathbf {\hat {x}} +\cos \theta \cos \varphi \mathbf {\hat {y}} =\cos \theta {\boldsymbol {\hat {\varphi }}}}
∂θ^∂θ=−sin⁡θcos⁡φx^−sin⁡θsin⁡φy^−cos⁡θz^=−r^{\displaystyle {\frac {\partial {\boldsymbol {\hat {\theta }}}}{\partial \theta }}=-\sin \theta \cos \varphi \mathbf {\hat {x}} -\sin \theta \sin \varphi \mathbf {\hat {y}} -\cos \theta \mathbf {\hat {z}} =-\mathbf {\hat {r}} }
∂φ^∂φ=−cos⁡φx^−sin⁡φy^=−sin⁡θr^−cos⁡θθ^{\displaystyle {\frac {\partial {\boldsymbol {\hat {\varphi }}}}{\partial \varphi }}=-\cos \varphi \mathbf {\hat {x}} -\sin \varphi \mathbf {\hat {y}} =-\sin \theta \mathbf {\hat {r}} -\cos \theta {\boldsymbol {\hat {\theta }}}}

General unit vectors

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Main article:Orthogonal coordinates

Common themes of unit vectors occur throughoutphysics andgeometry:[4]

Unit vectorNomenclatureDiagram
Tangent vector to a curve/flux linet^{\displaystyle \mathbf {\hat {t}} }"200px""200px"

A normal vectorn^{\displaystyle \mathbf {\hat {n}} } to the plane containing and defined by the radial position vectorrr^{\displaystyle r\mathbf {\hat {r}} } and angular tangential direction of rotationθθ^{\displaystyle \theta {\boldsymbol {\hat {\theta }}}} is necessary so that the vector equations of angular motion hold.

Normal to a surface tangent plane/plane containing radial position component and angular tangential componentn^{\displaystyle \mathbf {\hat {n}} }

In terms ofpolar coordinates;n^=r^×θ^{\displaystyle \mathbf {\hat {n}} =\mathbf {\hat {r}} \times {\boldsymbol {\hat {\theta }}}}

Binormal vector to tangent and normalb^=t^×n^{\displaystyle \mathbf {\hat {b}} =\mathbf {\hat {t}} \times \mathbf {\hat {n}} }[5]
Parallel to some axis/linee^∥{\displaystyle \mathbf {\hat {e}} _{\parallel }}"200px"

One unit vectore^∥{\displaystyle \mathbf {\hat {e}} _{\parallel }} aligned parallel to a principal direction (red line), and a perpendicular unit vectore^⊥{\displaystyle \mathbf {\hat {e}} _{\bot }} is in any radial direction relative to the principal line.

Perpendicular to some axis/line in some radial directione^⊥{\displaystyle \mathbf {\hat {e}} _{\bot }}
Possible angular deviation relative to some axis/linee^∠{\displaystyle \mathbf {\hat {e}} _{\angle }}"200px"

Unit vector at acute deviation angleφ (including 0 orπ/2 rad) relative to a principal direction.

Curvilinear coordinates

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In general, a coordinate system may be uniquely specified using a number oflinearly independent unit vectorse^n{\displaystyle \mathbf {\hat {e}} _{n}}[1] (the actual number being equal to the degrees of freedom of the space). For ordinary 3-space, these vectors may be denotede^1,e^2,e^3{\displaystyle \mathbf {\hat {e}} _{1},\mathbf {\hat {e}} _{2},\mathbf {\hat {e}} _{3}}. It is nearly always convenient to define the system to be orthonormal andright-handed:

e^i⋅e^j=δij{\displaystyle \mathbf {\hat {e}} _{i}\cdot \mathbf {\hat {e}} _{j}=\delta _{ij}}
e^i⋅(e^j×e^k)=εijk{\displaystyle \mathbf {\hat {e}} _{i}\cdot (\mathbf {\hat {e}} _{j}\times \mathbf {\hat {e}} _{k})=\varepsilon _{ijk}}

whereδij{\displaystyle \delta _{ij}} is theKronecker delta (which is 1 fori =j, and 0 otherwise) andεijk{\displaystyle \varepsilon _{ijk}} is theLevi-Civita symbol (which is 1 for permutations ordered asijk, and −1 for permutations ordered askji).

Right versor

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A unit vector inR3{\displaystyle \mathbb {R} ^{3}} was called aright versor byW. R. Hamilton, as he developed hisquaternionsH⊂R4{\displaystyle \mathbb {H} \subset \mathbb {R} ^{4}}. In fact, he was the originator of the termvector, as every quaternionq=s+v{\displaystyle q=s+v} has a scalar parts and a vector partv. Ifv is a unit vector inR3{\displaystyle \mathbb {R} ^{3}}, then the square ofv in quaternions is −1. Thus byEuler's formula,exp⁡(θv)=cos⁡θ+vsin⁡θ{\displaystyle \exp(\theta v)=\cos \theta +v\sin \theta } is aversor in the3-sphere. Whenθ is aright angle, the versor is a right versor: its scalar part is zero and its vector partv is a unit vector inR3{\displaystyle \mathbb {R} ^{3}}.

Thus the right versors extend the notion ofimaginary units found in thecomplex plane, where the right versors now range over the2-sphereS2⊂R3⊂H{\displaystyle \mathbb {S} ^{2}\subset \mathbb {R} ^{3}\subset \mathbb {H} } rather than the pair{i, −i} in the complex plane.

By extension, aright quaternion is a real multiple of a right versor.

See also

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Look upunit vector in Wiktionary, the free dictionary.

Notes

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  1. ^abcWeisstein, Eric W."Unit Vector".Wolfram MathWorld. Retrieved2020-08-19.
  2. ^"Unit Vectors".Brilliant Math & Science Wiki. Retrieved2020-08-19.
  3. ^Tevian Dray and Corinne A. Manogue, Spherical Coordinates, College Math Journal 34, 168-169 (2003).
  4. ^F. Ayres; E. Mendelson (2009).Calculus (Schaum's Outlines Series) (5th ed.). Mc Graw Hill.ISBN 978-0-07-150861-2.
  5. ^M. R. Spiegel; S. Lipschutz; D. Spellman (2009).Vector Analysis (Schaum's Outlines Series) (2nd ed.). Mc Graw Hill.ISBN 978-0-07-161545-7.

References

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