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 …

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

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

Standard basis vectors

Three common unit vectors are $\hat{𝐢}$ , $\hat{𝐣}$ and $\hat{𝐤}$ , 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: $\hat{𝐢} = [\begin{matrix} 1 & 0 & 0 \end{matrix}], \hat{𝐣} = [\begin{matrix} 0 & 1 & 0 \end{matrix}], \hat{𝐤} = [\begin{matrix} 0 & 0 & 1 \end{matrix}]$

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

Euclidean vector

References

↑ "Comprehensive List of Algebra Symbols". Math Vault. 2020-03-25. Retrieved 2020-08-19.
↑ "Unit Vector". www.mathsisfun.com. Retrieved 2020-08-19.
↑ Weisstein, Eric W. "Unit Vector". mathworld.wolfram.com. Retrieved 2020-08-19.
↑ ^4.0 ^4.1 "Unit Vectors | Brilliant Math & Science Wiki". brilliant.org. Retrieved 2020-08-19.

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Unit

Rabu, 2025-09-24 07:17:12

Unit means part of something: a ~ in an army/air force, a navy. Component, building block of constructions and molecules. It could mean a unit of measurement...

Unit vector

Standard basis vectors

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