# Anti-Commutative Vector Cross Product Vectors anti-commute in euclidean space. The anti-commutative property is when swapping the two terms in an operation results in a negated result. $ (\vec a \times \vec b) = -(\vec b \times \vec a) $ This means that a [[Vector Matrix Multiplication#Cross Product|cross product]] with themselves (or another equivalent vector) always equals zero. $ \vec a \times \vec a = 0 $ As a result, a vector in a euclidean space taken to an even power using a cross product is zero. $ \vec a^{n \ \text{is even}} = 0 $ This also means that the [[Vector Matrix Multiplication#Geometric Product|geometric product]] ($*$) of a vector with itself is equal to just the [[Vector Matrix Multiplication#Dot Product|dot product]] because the [[Vector Matrix Multiplication#Geometric Product#Quaternions|wedge product]] component is zeroed out. $ \vec a * \vec a = \vec a \cdot \vec a $ The result of an even number of operations is a *scalar*. $ \vec a * \vec a = x $ While the result of an odd number of geometric products is a *vector* because it becomes a scalar multiplying against a vector. This sounds nonsensical until you realize that the scalar is actually a quaternion result with empty elements. $ \vec a * \vec a * \vec a = \vec v $ Finally, when taken all together, this means that whenever vectors anti-commute, only their dot product matters when squaring or taking an even power. $ \begin{gathered} (\vec a \times \vec b) = -(\vec b \times \vec a) \\ \therefore \\ \vec a \times \vec a = 0 \\ \therefore \\ \vec a * \vec a = \vec a \cdot \vec a = x \\ \therefore \\ \vec a^{n \ \text{is even}} = x \\ \vec a^{n \ \text{is odd}} = \vec v \end{gathered} $ # Dot Product and Perpendicular Vectors If the [[Vector Matrix Multiplication#Dot Product|dot product]] of two vectors is zero then they are perpendicular. Likewise, if two vectors are perpendicular their dot product is always zero. $ \vec a \perp \vec b \iff \vec a \cdot \vec b = 0 $ # Dot Product and Magnitude The [[Vector Matrix Multiplication#Dot Product|dot product]] of a vector with a [vector of ones](https://en.wikipedia.org/w/index.php?title=Matrix_of_ones) $J$ with the same cardinality $n$ is equal to the magnitude of the initial vector. $\begin{gathered} n = \# \vec a \\ \vec a \cdot \vec J_n = \lvert\lvert \vec a \rvert\rvert = m \end{gathered}$ # Wedge Product and Curvature To obtain the curvature in any dimension you can apply the [[Vector Matrix Multiplication#Wedge Product|wedge product]] over the magnitude. $ k = \frac {\lvert\lvert f' \wedge f'' \rvert\rvert} {\lvert\lvert f' \rvert\rvert^3} $ $k$ is the curvature, also written as $k(t)$. The denominator is the magnitude of the vector cubed. The numerator is the wedge product of the first and second differential. **TODO**: explain the source of the functions and their relationship to the curve. # Wedge Product and Identity A wedge product of two vectors will product a higher order vector perpendicular to both. If both vectors are equal, then the result will be a higher order [[Math Notation Mathtex and LaTeX#Zero Elements|zero vector]]. # Geometric Product and Rotation [[Quaternion]]s are equivalent to [[Vector Geometric Algebra#Rotor|rotors]]. Both are used for rotation in 2D and 3D space. For normalized input vectors the output of the vector multiplication will give us a quaternion rotor with a magnitude of `1` (a unit quaternion). If that rotor is applied as a rotation then it will result in a rotation of double the angular difference between the two input vectors. $ \angle \vec a - \angle \vec b = 2\theta $ # Normalizing a Vector > > - A normal vector of length one is called a unit normal vector. > - A curvature vector is a normal vector whose length is the curvature of the object. > - Multiplying a normal vector by -1 results in the opposite vector, which may be used for indicating sides (e.g., interior or exterior). > The unit vector of a vector A is the vector with the same initial point and direction as A, but with a length of 1 unit. # Magnitude vs Length Magnitude and length are equivalent terms. # Other Topics to Cover Vector Derivatives, vector's relationship to differential geometry, # References - https://www.euclideanspace.com/maths/algebra/vectors/vecAlgebra/powers/index.htm - https://math.libretexts.org/Bookshelves/Calculus/Calculus_(OpenStax)/12%3A_Vectors_in_Space/12.03%3A_The_Dot_Product - https://stackoverflow.com/questions/10002918/what-is-the-need-for-normalizing-a-vector - https://www.wikihow.com/Normalize-a-Vector - https://math.stackexchange.com/questions/643227/in-the-context-of-vectors-is-there-a-difference-between-the-terms-magnitude-a - https://onlinemschool.com/math/library/vector/length/ - https://math.stackexchange.com/questions/3527712/proving-that-kt-frac-alpha-wedge-alpha-alpha3 - https://math.stackexchange.com/questions/1457699/easy-derivation-to-show-that-the-curvature-is-kt-frac-alpha-wedge-alpha/1463306#1463306 - https://en.wikipedia.org/wiki/Vector_calculus - https://en.wikipedia.org/wiki/Derivative