Unit vector

A unit vector is any vector that is one unit in length. Unit vectors are often notated the same way as normal vectors, but with a mark called a circumflex over the letter (e.g. [math]\displaystyle{ \mathbf{\hat{v}} }[/math] is the unit vector of [math]\displaystyle{ \mathbf{v} }[/math].)

To make a vector into a unit vector, one just needs to divide it by its length: [math]\displaystyle{ \hat{\mathbf{v}} = \mathbf{v} / \lVert \mathbf{v} \rVert }[/math]. The resulting unit vector will be in the same direction as the original vector.

Standard basis vectors

Three common unit vectors are [math]\displaystyle{ \mathbf{\hat{i}} }[/math], [math]\displaystyle{ \mathbf{\hat{j}} }[/math] and [math]\displaystyle{ \mathbf{\hat{k}} }[/math], referring to the three-dimensional unit vectors for the x-, y- and z-axes, respectively. These vectors are called the standard basis vectors of a 3-dimensional Cartesian coordinate system. They are commonly just notated as i, j and k.

They can be written as follows: [math]\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} }[/math]

For the [math]\displaystyle{ i }[/math]-th standard basis vector of a vector space, the symbol [math]\displaystyle{ e_i }[/math] (or [math]\displaystyle{ \hat{e}_i }[/math]) may be used. This refers to the vector with 1 in the [math]\displaystyle{ i }[/math]-th component, and 0 elsewhere.

Unit Vector Media

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  Examples of two 2D direction vectors

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  Examples of two 3D direction vectors

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  unit vectors in a right hand triad, tangent to a given curve, normal to the surface mesh, binormal to another curve

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  how the unit vectors in polar coordinates relate to the normal in the plane defined by the unit vectors themselves

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  Perpendicular and parallel unit vectors

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  unit vector deviated at an angle relative to a principle direction

Related pages

• Euclidean vector