In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) whose length is 1. A unit vector is often written with a superscribed caret or “hat”, thus: $\{\backslash hat\{i\}\}$ (pronounced "i-hat").

In Euclidean space, the dot product of two unit vectors is simply the cosine of the angle between them. This follows from the formula for the dot product, since the lengths are both 1.

The normalized vector $\{\backslash hat\{u\}\}$ of a non-zero vector u is the unit vector codirectional with u, i.e.,

$\backslash mathbf\{\backslash hat\{u\}\}\; =\; \backslash frac\{\backslash mathbf\{u\}\}\{\backslash |\backslash mathbf\{u\}\backslash |\}.$

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

The elements of a basis are usually chosen to be unit vectors. Every vector in the space may be written as a linear combination of unit vectors, with the components of each being given by direction cosines. The most commonly encountered bases are Cartesian, polar, and spherical coordinates. Each uses different unit vectors according to the symmetry of the coordinate system. Since these systems are encountered in so many different contexts, it is not uncommon to encounter different naming conventions than those used here. Usually, a little context should enable the astute reader to substitute the names being used for those given here.

Cartesian Coordinates

In the 3-Dimensional Cartesian coordinate system, the unit vectors along the x, y, and z axes are usually denoted i, j, and k, respectively.

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

These are sometimes written using normal vector notation rather than the hat/caret notation, and it can generally be assumed that $\backslash vec\{i\},\; \backslash vec\{j\},\; \backslash vec\{k\}$ are unit vectors in most contexts. The notation $\backslash hat\{x\},\; \backslash hat\{y\},\; \backslash hat\{z\}$ is also used, particularly in contexts where i, j, or k might lead to confusion with another quantity.

Cylindrical Coordinates

The unit vectors appropriate to cylindrical symmetry are: $\backslash boldsymbol\{\backslash hat\{s\}\}$, the distance from the axis of symmetry; $\backslash boldsymbol\{\backslash hat\; \backslash phi\}$, the angle measured counterclockwise from the positive x-axis; and $\backslash boldsymbol\{\backslash hat\{z\}\}$. They are related to the Cartesian basis $\backslash hat\{x\},\; \backslash hat\{y\},\; \backslash hat\{z\}$ by:

$\backslash boldsymbol\{\backslash hat\{s\}\}$ = $\backslash cos\; \backslash phi\backslash boldsymbol\{\backslash hat\{x\}\}\; +\; \backslash sin\; \backslash phi\backslash boldsymbol\{\backslash hat\{y\}\}$

$\backslash boldsymbol\{\backslash hat\; \backslash phi\}$ = $-\backslash sin\; \backslash phi\backslash boldsymbol\{\backslash hat\{x\}\}\; +\; \backslash cos\; \backslash phi\backslash boldsymbol\{\backslash hat\{y\}\}$

$\backslash boldsymbol\{\backslash hat\{z\}\}=\backslash boldsymbol\{\backslash hat\{z\}\}$

It is important to note that $\backslash boldsymbol\{\backslash hat\{s\}\}$ and $\backslash boldsymbol\{\backslash hat\; \backslash phi\}$ are functions of $\backslash phi$, 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. The derivatives with respect to $\backslash phi$ are:

$\backslash frac\{\backslash partial\; \backslash boldsymbol\{\backslash hat\{s\}\}\}\; \{\backslash partial\; \backslash phi\}\; =\; -\backslash sin\; \backslash phi\backslash boldsymbol\{\backslash hat\{x\}\}\; +\; \backslash cos\; \backslash phi\backslash boldsymbol\{\backslash hat\{y\}\}\; =\; \backslash boldsymbol\{\backslash hat\; \backslash phi\}$

$\backslash frac\{\backslash partial\; \backslash boldsymbol\{\backslash hat\; \backslash phi\}\}\; \{\backslash partial\; \backslash phi\}\; =\; -\backslash cos\; \backslash phi\backslash boldsymbol\{\backslash hat\{x\}\}\; -\; \backslash sin\; \backslash phi\backslash boldsymbol\{\backslash hat\{y\}\}\; =\; -\backslash boldsymbol\{\backslash hat\{s\}\}$

$\backslash frac\{\backslash partial\; \backslash boldsymbol\{\backslash hat\{z\}\}\}\; \{\backslash partial\; \backslash phi\}\; =\; 0$

Spherical Coordinates

The unit vectors appropriate to spherical symmetry are: $\backslash boldsymbol\{\backslash hat\{r\}\}$, the radial distance from the origin; $\backslash boldsymbol\{\backslash hat\{\backslash phi\}\}$, the angle in the x-y plane counterclockwise from the positive x-axis; and $\backslash boldsymbol\{\backslash hat\; \backslash theta\}$, the angle from the positive z axis. To minimize degeneracy, the polar angle is usually taken $0\backslash leq\backslash theta\backslash leq\; 180^\backslash circ$. It is especially important to note the context of any ordered triplet written in spherical coordinates, as the roles of $\backslash boldsymbol\{\backslash hat\; \backslash phi\}$ and $\backslash boldsymbol\{\backslash hat\; \backslash theta\}$ are often reversed. Here, the American naming convention is used. This leaves the azimuthal angle $\backslash phi$ defined the same as in cylindrical coordinates. The Cartesian relations are:

$\backslash boldsymbol\{\backslash hat\{r\}\}\; =\; \backslash sin\; \backslash theta\; \backslash cos\; \backslash phi\backslash boldsymbol\{\backslash hat\{x\}\}\; +\; \backslash sin\; \backslash theta\; \backslash sin\; \backslash phi\backslash boldsymbol\{\backslash hat\{y\}\}\; +\; \backslash cos\; \backslash theta\backslash boldsymbol\{\backslash hat\{z\}\}$

$\backslash boldsymbol\{\backslash hat\; \backslash theta\}\; =\; \backslash cos\; \backslash theta\; \backslash cos\; \backslash phi\backslash boldsymbol\{\backslash hat\{x\}\}\; +\; \backslash cos\; \backslash theta\; \backslash sin\; \backslash phi\backslash boldsymbol\{\backslash hat\{y\}\}\; -\; \backslash sin\; \backslash theta\backslash boldsymbol\{\backslash hat\{z\}\}$

$\backslash boldsymbol\{\backslash hat\; \backslash phi\}\; =\; -\backslash sin\; \backslash phi\backslash boldsymbol\{\backslash hat\{x\}\}\; +\; \backslash cos\; \backslash phi\backslash boldsymbol\{\backslash hat\{y\}\}$

The spherical unit vectors depend on both $\backslash phi$ and $\backslash theta$, and hence there are 5 possible non-zero derivates. For a more complete description, see Jacobian. The non-zero derivatives are:

$\backslash frac\{\backslash partial\; \backslash boldsymbol\{\backslash hat\{r\}\}\}\; \{\backslash partial\; \backslash phi\}\; =\; -\backslash sin\; \backslash theta\; \backslash sin\; \backslash phi\backslash boldsymbol\{\backslash hat\{x\}\}\; +\; \backslash sin\; \backslash theta\; \backslash cos\; \backslash phi\backslash boldsymbol\{\backslash hat\{y\}\}\; =\; \backslash sin\; \backslash theta\backslash boldsymbol\{\backslash hat\; \backslash phi\}$

$\backslash frac\{\backslash partial\; \backslash boldsymbol\{\backslash hat\{r\}\}\}\; \{\backslash partial\; \backslash theta\}\; =\backslash cos\; \backslash theta\; \backslash cos\; \backslash phi\backslash boldsymbol\{\backslash hat\{x\}\}\; +\; \backslash cos\; \backslash theta\; \backslash sin\; \backslash phi\backslash boldsymbol\{\backslash hat\{y\}\}\; -\; \backslash sin\; \backslash theta\backslash boldsymbol\{\backslash hat\{z\}\}=\; \backslash boldsymbol\{\backslash hat\; \backslash theta\}$

$\backslash frac\{\backslash partial\; \backslash boldsymbol\{\backslash hat\{\backslash theta\}\}\}\; \{\backslash partial\; \backslash phi\}\; =-\backslash cos\; \backslash theta\; \backslash sin\; \backslash phi\backslash boldsymbol\{\backslash hat\{x\}\}\; +\; \backslash cos\; \backslash theta\; \backslash cos\; \backslash phi\backslash boldsymbol\{\backslash hat\{y\}\}\; =\; \backslash cos\; \backslash theta\backslash boldsymbol\{\backslash hat\; \backslash phi\}$

$\backslash frac\{\backslash partial\; \backslash boldsymbol\{\backslash hat\{\backslash theta\}\}\}\; \{\backslash partial\; \backslash theta\}\; =\; -\backslash sin\; \backslash theta\; \backslash cos\; \backslash phi\backslash boldsymbol\{\backslash hat\{x\}\}\; -\; \backslash sin\; \backslash theta\; \backslash sin\; \backslash phi\backslash boldsymbol\{\backslash hat\{y\}\}\; -\; \backslash cos\; \backslash theta\backslash boldsymbol\{\backslash hat\{z\}\}\; =\; -\backslash boldsymbol\{\backslash hat\{r\}\}$

$\backslash frac\{\backslash partial\; \backslash boldsymbol\{\backslash hat\{\backslash phi\}\}\}\; \{\backslash partial\; \backslash phi\}\; =\; -\backslash cos\; \backslash phi\backslash boldsymbol\{\backslash hat\{x\}\}\; +\; \backslash sin\; \backslash phi\backslash boldsymbol\{\backslash hat\{y\}\}$

Further reading

• Cartesian coordinate system
• Polar coordinate system
• Coordinates (mathematics)
• Curvilinear coordinates
• Jacobian

References

• Mathematical Methods for Physicists G. B. Arfken & H. J. Weber, 5^th ed.
• Schaum's Outlines: Mathematical Handbook of Formulas and Tables
• Introduction to Electrodynamics David J. Griffiths, 3^th ed.

Linear algebra | Elementary mathematics

Enhedsvektor | Einheitsvektor | Vecteur unitaire | וקטור יחידה | Eenheidsvector | Wersor | Vetor unitário | Enhetsvektor | เวกเตอร์หนึ่งหน่วย | 单位向量