Unit vector

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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. ${\displaystyle \mathbf {\hat {v}} }$ is the unit vector of ${\displaystyle \mathbf {v} }$.)[1][2]

To make a vector into a unit vector, one just needs to divide it by its length: ${\displaystyle {\hat {\mathbf {v} }}=\mathbf {v} /\lVert \mathbf {v} \rVert }$.[3] The resulting unit vector will be in the same direction as the original vector.[4]

Standard basis vectors

Three common unit vectors are ${\displaystyle \mathbf {\hat {i}} }$, ${\displaystyle \mathbf {\hat {j}} }$ and ${\displaystyle \mathbf {\hat {k}} }$, 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: ${\displaystyle \mathbf {\hat {i}} ={\begin{bmatrix}1&0&0\end{bmatrix}},\,\,\mathbf {\hat {j}} ={\begin{bmatrix}0&1&0\end{bmatrix}},\,\,\mathbf {\hat {k}} ={\begin{bmatrix}0&0&1\end{bmatrix}}}$

For the ${\displaystyle i}$-th standard basis vector of a vector space, the symbol ${\displaystyle e_{i}}$ (or ${\displaystyle {\hat {e}}_{i}}$) may be used.[4] This refers to the vector with 1 in the ${\displaystyle i}$-th component, and 0 elsewhere.

References

1. "Comprehensive List of Algebra Symbols". Math Vault. 2020-03-25. Retrieved 2020-08-19.
2. "Unit Vector". www.mathsisfun.com. Retrieved 2020-08-19.
3. Weisstein, Eric W. "Unit Vector". mathworld.wolfram.com. Retrieved 2020-08-19.
4. "Unit Vectors | Brilliant Math & Science Wiki". brilliant.org. Retrieved 2020-08-19.