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# Unit vector

In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat": ${\displaystyle {\hat {\imath }}}$ (pronounced "i-hat"). The term direction vector is used to describe a unit vector being used to represent spatial direction, and such quantities are commonly denoted as d. Two 2D direction vectors, d1 and d2 are illustrated. 2D spatial directions represented this way are numerically equivalent to points on the unit circle.

The same construct is used to specify spatial directions in 3D. As illustrated, each unique direction is equivalent numerically to a point on the unit sphere.

Examples of two 2D direction vectors
Examples of two 3D direction vectors

The normalized vector or versor û of a non-zero vector u is the unit vector in the direction of u, i.e.,

${\displaystyle \mathbf {\hat {u}} ={\frac {\mathbf {u} }{|\mathbf {u} |}}}$

where |u| is the norm (or length) of u. The term normalized vector is sometimes used as a synonym for unit vector.

Unit vectors are often chosen to form the basis of a vector space. Every vector in the space may be written as a linear combination of unit vectors.

By definition, in a Euclidean space the dot product of two unit vectors is a scalar value amounting to the cosine of the smaller subtended angle. In three-dimensional Euclidean space, the cross product of two arbitrary unit vectors is a third vector orthogonal to both of them having length equal to the sine of the smaller subtended angle. The normalized cross product corrects for this varying length, and yields the mutually orthogonal unit vector to the two inputs, applying the right-hand rule to resolve one of two possible directions.

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#### Transcription

We've already seen that you can visually represent a vector as an arrow, where the length of the arrow is the magnitude of the vector and the direction of the arrow is the direction of the vector. And if we want to represent this mathematically, we could just think about, well, starting from the tail of the vector, how far away is the head of the vector in the horizontal direction? And how far away is it in the vertical direction? So for example, in the horizontal direction, you would have to go this distance. And then in the vertical direction, you would have to go this distance. Let me do that in a different color. You would have to go this distance right over here. And so let's just say that this distance is 2 and that this distance is 3. We could represent this vector-- and let's call this vector v. We could represent vector v as an ordered list or a 2-tuple of-- so we could say we move 2 in the horizontal direction and 3 in the vertical direction. So you could represent it like that. You could represent vector v like this, where it is 2 comma 3, like that. And what I now want to introduce you to-- and we could come up with other ways of representing this 2-tuple-- is another notation. And this really comes out of the idea of what it means to add and scale vectors. And to do that, we're going to define what we call unit vectors. And if we're in two dimensions, we define a unit vector for each of the dimensions we're operating in. If we're in three dimensions, we would define a unit vector for each of the three dimensions that we're operating in. And so let's do that. So let's define a unit vector i. And the way that we denote that is the unit vector is, instead of putting an arrow on top, we put this hat on top of it. So the unit vector i, if we wanted to write it in this notation right over here, we would say it only goes 1 unit in the horizontal direction, and it doesn't go at all in the vertical direction. So it would look something like this. That is the unit vector i. And then we can define another unit vector. And let's call that unit vector-- or it's typically called j, which would go only in the vertical direction and not in the horizontal direction. And not in the horizontal direction, and it goes 1 unit in the vertical direction. So this went 1 unit in the horizontal. And now j is going to go 1 unit in the vertical. So j-- just like that. Now any vector, any two dimensional vector, we can now represent as a sum of scaled up versions of i and j. And you say, well, how do we do that? Well, you could imagine vector v right here is the sum of a vector that moves purely in the horizontal direction that has a length 2, and a vector that moves purely in the vertical direction that has length 3. So we could say that vector v-- let me do it in that same blue color-- is equal to-- so if we want a vector that has length 2 and it moves purely in the horizontal direction, well, we could just scale up the unit vector i. We could just multiply 2 times i. So let's do that-- is equal to 2 times our unit vector i. So 2i is going to be this whole thing right over here or this whole vector. Let me do it in this yellow color. This vector right over here, you could view as 2i. And then to that, we're going to add 3 times j-- so plus 3 times j. Let me write it like this. Let me get that color. Once again, 3 times j is going to be this vector right over here. And if you add this yellow vector right over here to the magenta vector, you're going to get-- notice, we're putting the tail of the magenta vector at the head of the yellow vector. And if you start at the tail of the yellow vector and you go all the way to the head of the magenta vector, you have now constructed vector v. So vector v, you could represent it as a column vector like this, 2 3. You could represent it as 2 comma 3, or you could represent it as 2 times i with this little hat over it, plus 3 times j, with this little hat over it. i is the unit vector in the horizontal direction, in the positive horizontal direction. If you want to go the other way, you would multiply it by a negative. And j is the unit vector in the vertical direction. As we'll see in future videos, once you go to three dimensions, you'll introduce a k. But it's very natural to translate between these two things. Notice, 2, 3-- 2, 3. And so with that, let's actually do some vector operations using this notation. So let's say that I define another vector. Let's say it is vector b. I'll just come up with some numbers here. Vector b is equal to negative 1 times i-- times the unit vector i-- plus 4 times the unit vector in the horizontal direction. So given these two vector definitions, what would the would be the vector v plus b be equal to? And I encourage you to pause the video and think about it. Well once again, we just literally have to add corresponding components. We could say, OK, well let's think about what we're doing in the horizontal direction. We're going 2 in the horizontal direction here, and now we're going negative 1. So our horizontal component is going to be 2 plus negative 1-- 2 plus negative 1 in the horizontal direction. And we're going to multiply that times the unit vector i. And this, once again, just goes back to adding the corresponding components of the vector. And then we're going to have plus 4, or plus 3 plus 4-- And let me write it that way-- times the unit vector j in the vertical direction. And so that's going to give us-- I'll do this all in this one color-- 2 plus negative 1 is 1i. And we could literally write that just as i. Actually, let's do that. Let's just write that as i. But we got that from 2 plus negative 1 is 1. 1 times the vector is just going to be that vector, plus 3 plus 4 is 7-- 7j. And you see, this is exactly how we saw vector addition in the past, is that we could also represent vector b like this. We could represent it like this-- negative 1, 4. And so if you were to add v to b, you add the corresponding terms. So if we were to add corresponding terms, looking at them as column vectors, that is going to be equal to 2 plus negative 1, which is 1. 3 plus 4 is 7. So this is the exact same representation as this. This is using unit vector notation, and this is representing it as a column vector.

## Orthogonal coordinates

### Cartesian coordinates

Unit vectors may be used to represent the axes of a Cartesian coordinate system. For instance, the unit vectors in the direction of the x, y, and z axes of a three dimensional Cartesian coordinate system are

${\displaystyle \mathbf {\hat {i}} ={\begin{bmatrix}1\\0\\0\end{bmatrix}},\,\,\mathbf {\hat {j}} ={\begin{bmatrix}0\\1\\0\end{bmatrix}},\,\,\mathbf {\hat {k}} ={\begin{bmatrix}0\\0\\1\end{bmatrix}}}$

They are sometimes referred to as the versors of the coordinate system, and they form a set of mutually orthogonal unit vectors, typically referred to as a standard basis in linear algebra.

They are often denoted using normal vector notation (e.g., i or ${\displaystyle {\vec {\imath }}}$) rather than standard unit vector notation (e.g., ${\displaystyle \mathbf {\hat {\imath }} }$). In most contexts it can be assumed that i, j, and k, (or ${\displaystyle {\vec {\imath }},}$ ${\displaystyle {\vec {\jmath }},}$ and ${\displaystyle {\vec {k}}}$) are versors of a 3-D Cartesian coordinate system. The notations ${\displaystyle (\mathbf {\hat {x}} ,\mathbf {\hat {y}} ,\mathbf {\hat {z}} )}$, ${\displaystyle (\mathbf {\hat {x}} _{1},\mathbf {\hat {x}} _{2},\mathbf {\hat {x}} _{3})}$, ${\displaystyle (\mathbf {\hat {e}} _{x},\mathbf {\hat {e}} _{y},\mathbf {\hat {e}} _{z})}$, or ${\displaystyle (\mathbf {\hat {e}} _{1},\mathbf {\hat {e}} _{2},\mathbf {\hat {e}} _{3})}$, with or without hat, are also used, particularly in contexts where i, j, k might lead to confusion with another quantity (for instance with index symbols such as i, j, k, used to identify an element of a set or array or sequence of variables).

When a unit vector in space is expressed, with Cartesian notation, as a linear combination of i, j, k, its three scalar components can be referred to as direction 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 the orientation (angular position) of a straight line, segment of straight line, oriented axis, or segment of oriented axis (vector).

### Cylindrical coordinates

The three orthogonal unit vectors appropriate to cylindrical symmetry are:

• ${\displaystyle \mathbf {\hat {\rho }} }$ (also designated ${\displaystyle \mathbf {\hat {e}} }$ or ${\displaystyle {\boldsymbol {\hat {s}}}}$), representing the direction along which the distance of the point from the axis of symmetry is measured;
• ${\displaystyle {\boldsymbol {\hat {\varphi }}}}$, representing the direction of the motion that would be observed if the point were rotating counterclockwise about the symmetry axis;
• ${\displaystyle \mathbf {\hat {z}} }$, representing the direction of the symmetry axis;

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

${\displaystyle \mathbf {\hat {\rho }} }$ = ${\displaystyle \cos \varphi \mathbf {\hat {x}} +\sin \varphi \mathbf {\hat {y}} }$
${\displaystyle {\boldsymbol {\hat {\varphi }}}}$ = ${\displaystyle -\sin \varphi \mathbf {\hat {x}} +\cos \varphi \mathbf {\hat {y}} }$
${\displaystyle \mathbf {\hat {z}} =\mathbf {\hat {z}} .}$

It is important to note that ${\displaystyle \mathbf {\hat {\rho }} }$ and ${\displaystyle {\boldsymbol {\hat {\varphi }}}}$ are functions of ${\displaystyle \varphi }$, and are not constant in direction. When differentiating or integrating in cylindrical coordinates, these unit vectors themselves must also be operated on. For a more complete description, see Jacobian matrix. The derivatives with respect to ${\displaystyle \varphi }$ are:

${\displaystyle {\frac {\partial \mathbf {\hat {\rho }} }{\partial \varphi }}=-\sin \varphi \mathbf {\hat {x}} +\cos \varphi \mathbf {\hat {y}} ={\boldsymbol {\hat {\varphi }}}}$
${\displaystyle {\frac {\partial {\boldsymbol {\hat {\varphi }}}}{\partial \varphi }}=-\cos \varphi \mathbf {\hat {x}} -\sin \varphi \mathbf {\hat {y}} =-\mathbf {\hat {\rho }} }$
${\displaystyle {\frac {\partial \mathbf {\hat {z}} }{\partial \varphi }}=\mathbf {0} .}$

### Spherical coordinates

The unit vectors appropriate to spherical symmetry are: ${\displaystyle \mathbf {\hat {r}} }$, the direction in which the radial distance from the origin increases; ${\displaystyle {\boldsymbol {\hat {\varphi }}}}$, the direction in which the angle in the x-y plane counterclockwise from the positive x-axis is increasing; and ${\displaystyle {\boldsymbol {\hat {\theta }}}}$, the direction in which the angle from the positive z axis is increasing. To minimize redundancy of representations, the polar angle ${\displaystyle \theta }$ is usually taken to lie between zero and 180 degrees. It is especially important to note the context of any ordered triplet written in spherical coordinates, as the roles of ${\displaystyle {\boldsymbol {\hat {\varphi }}}}$ and ${\displaystyle {\boldsymbol {\hat {\theta }}}}$ are often reversed. Here, the American "physics" convention[1] is used. This leaves the azimuthal angle ${\displaystyle \varphi }$ defined the same as in cylindrical coordinates. The Cartesian relations are:

${\displaystyle \mathbf {\hat {r}} =\sin \theta \cos \varphi \mathbf {\hat {x}} +\sin \theta \sin \varphi \mathbf {\hat {y}} +\cos \theta \mathbf {\hat {z}} }$
${\displaystyle {\boldsymbol {\hat {\theta }}}=\cos \theta \cos \varphi \mathbf {\hat {x}} +\cos \theta \sin \varphi \mathbf {\hat {y}} -\sin \theta \mathbf {\hat {z}} }$
${\displaystyle {\boldsymbol {\hat {\varphi }}}=-\sin \varphi \mathbf {\hat {x}} +\cos \varphi \mathbf {\hat {y}} }$

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

${\displaystyle {\frac {\partial \mathbf {\hat {r}} }{\partial \varphi }}=-\sin \theta \sin \varphi \mathbf {\hat {x}} +\sin \theta \cos \varphi \mathbf {\hat {y}} =\sin \theta {\boldsymbol {\hat {\varphi }}}}$
${\displaystyle {\frac {\partial \mathbf {\hat {r}} }{\partial \theta }}=\cos \theta \cos \varphi \mathbf {\hat {x}} +\cos \theta \sin \varphi \mathbf {\hat {y}} -\sin \theta \mathbf {\hat {z}} ={\boldsymbol {\hat {\theta }}}}$
${\displaystyle {\frac {\partial {\boldsymbol {\hat {\theta }}}}{\partial \varphi }}=-\cos \theta \sin \varphi \mathbf {\hat {x}} +\cos \theta \cos \varphi \mathbf {\hat {y}} =\cos \theta {\boldsymbol {\hat {\varphi }}}}$
${\displaystyle {\frac {\partial {\boldsymbol {\hat {\theta }}}}{\partial \theta }}=-\sin \theta \cos \varphi \mathbf {\hat {x}} -\sin \theta \sin \varphi \mathbf {\hat {y}} -\cos \theta \mathbf {\hat {z}} =-\mathbf {\hat {r}} }$
${\displaystyle {\frac {\partial {\boldsymbol {\hat {\varphi }}}}{\partial \varphi }}=-\cos \varphi \mathbf {\hat {x}} -\sin \varphi \mathbf {\hat {y}} =-\sin \theta \mathbf {\hat {r}} -\cos \theta {\boldsymbol {\hat {\theta }}}}$

### General unit vectors

Common general themes of unit vectors occur throughout physics and geometry:[2]

Unit vector Nomenclature Diagram
Tangent vector to a curve/flux line ${\displaystyle \mathbf {\hat {t}} }$

A normal vector ${\displaystyle \mathbf {\hat {n}} }$ to the plane containing and defined by the radial position vector ${\displaystyle r\mathbf {\hat {r}} }$ and angular tangential direction of rotation ${\displaystyle \theta {\boldsymbol {\hat {\theta }}}}$ 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 component ${\displaystyle \mathbf {\hat {n}} }$

In terms of polar coordinates; ${\displaystyle \mathbf {\hat {n}} =\mathbf {\hat {r}} \times {\boldsymbol {\hat {\theta }}}}$

Binormal vector to tangent and normal ${\displaystyle \mathbf {\hat {b}} =\mathbf {\hat {t}} \times \mathbf {\hat {n}} }$[3]
Parallel to some axis/line ${\displaystyle \mathbf {\hat {e}} _{\parallel }}$

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

Perpendicular to some axis/line in some radial direction ${\displaystyle \mathbf {\hat {e}} _{\bot }}$
Possible angular deviation relative to some axis/line ${\displaystyle \mathbf {\hat {e}} _{\angle }}$

Unit vector at acute deviation angle φ (including 0 or π/2 rad) relative to a principal direction.

## Curvilinear coordinates

In general, a coordinate system may be uniquely specified using a number of linearly independent unit vectors ${\displaystyle \mathbf {\hat {e}} _{n}}$ equal to the degrees of freedom of the space. For ordinary 3-space, these vectors may be denoted ${\displaystyle \mathbf {\hat {e}} _{1},\mathbf {\hat {e}} _{2},\mathbf {\hat {e}} _{3}}$. It is nearly always convenient to define the system to be orthonormal and right-handed:

${\displaystyle \mathbf {\hat {e}} _{i}\cdot \mathbf {\hat {e}} _{j}=\delta _{ij}}$
${\displaystyle \mathbf {\hat {e}} _{i}\cdot (\mathbf {\hat {e}} _{j}\times \mathbf {\hat {e}} _{k})=\varepsilon _{ijk}}$

where ${\displaystyle \delta _{ij}}$ is the Kronecker delta (which is 1 for i = j and 0 otherwise) and ${\displaystyle \varepsilon _{ijk}}$ is the Levi-Civita symbol (which is 1 for permutations ordered as ijk and −1 for permutations ordered as kji).

## Right versor

A unit vector in ℝ3 was called a right versor by W. R. Hamilton as he developed his quaternions ℍ ⊂ ℝ4. In fact, he was the originator of the term vector as every quaternion ${\displaystyle q=s+v}$ has a scalar part s and a vector part v. If v is a unit vector in ℝ3, then the square of v in quaternions is –1. By Euler's formula then, ${\displaystyle \exp(\theta v)=\cos \theta +v\sin \theta }$ is a versor in the 3-sphere. When θ is a right angle, the versor is a right versor: its scalar part is zero and its vector part v is a unit vector in ℝ3.

## Notes

1. ^ Tevian Dray and Corinne A. Manogue,Spherical Coordinates, College Math Journal 34, 168-169 (2003).
2. ^ F. Ayres; E. Mandelson (2009). Calculus (Schaum's Outlines Series) (5th ed.). Mc Graw Hill. ISBN 978-0-07-150861-2.
3. ^ 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

• G. B. Arfken & H. J. Weber (2000). Mathematical Methods for Physicists (5th ed.). Academic Press. ISBN 0-12-059825-6.
• Spiegel, Murray R. (1998). Schaum's Outlines: Mathematical Handbook of Formulas and Tables (2nd ed.). McGraw-Hill. ISBN 0-07-038203-4.
• Griffiths, David J. (1998). Introduction to Electrodynamics (3rd ed.). Prentice Hall. ISBN 0-13-805326-X.