Normalized Vector
A Normalized Vector is a Unit Vector that results from dividing a vector by its norm.
- Context:
- It can be defined as : $\hat{\mathbf{X}}=\dfrac{\mathbf{X}}{\parallel\mathbf{X}\parallel}$
- Example(s):
- If $\mathbf{X}=(1,0,1)$ then the normalized vector $\hat{\mathbf{X}}=\left(1/\sqrt{2},0,1/\sqrt{2}\right)$
- Counter-Example(s):
- a Zero Vector.
- See: Vector Normalization Function, Cross Product, Normed Vector Space, Vector Space, Vector (Geometry), Euclidean Space, Dot Product.
References
2021
- (MathWorld,2021) ⇒ Eric W. Weisstein (2021). https://mathworld.wolfram.com/NormalizedVector.html Retrieved:2021-05-23.
- QUOTE: The normalized vector of $\mathbf{X}$ is a vector in the same direction but with norm (length) 1. It is denoted $\mathbf{\hat{X}}$ and given by$\mathbf{\hat{X}}=\dfrac{\mathbf{X}}{|\mathbf{X}|}$,
where $|\mathbf{X}|$ is the norm of $\mathbf{X}$. It is also called a unit vector.
- QUOTE: The normalized vector of $\mathbf{X}$ is a vector in the same direction but with norm (length) 1. It is denoted $\mathbf{\hat{X}}$ and given by
2015
- (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/unit_vector Retrieved:2015-2-7.
- 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 "hat": [math]\displaystyle{ {\hat{\imath}} }[/math] (pronounced "i-hat").
The normalized vector or versor [math]\displaystyle{ \mathbf{\hat{u}} }[/math] of a non-zero vector u is the unit vector in the direction of u, i.e.,
:[math]\displaystyle{ \mathbf{\hat{u}} = \frac{\mathbf{u}}{\|\mathbf{u}\|} }[/math]
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 an Euclidean space the dot product of two unit vectors is the cosine of the angle between them. In three-dimensional Euclidean space, the cross product of two orthogonal unit vectors is another unit vector, orthogonal to both of them.
- 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 "hat": [math]\displaystyle{ {\hat{\imath}} }[/math] (pronounced "i-hat").