## Euclidean vector

• In mathematics, physics, and engineering, a Euclidean vector (sometimes called a geometric or spatial vector, or—as here—simply a vector) is a geometric object that has magnitude (or length) and direction. Vectors can be added to other vectors according to vector algebra. A Euclidean vector is frequently represented by a line segment with a definite direction, or graphically as an arrow, connecting an initial point A with a terminal point B, and denoted by ${\displaystyle {\overrightarrow {AB}}.}$

A vector is what is needed to "carry" the point A to the point B; the Latin word vector means "carrier". It was first used by 18th century astronomers investigating planet rotation around the Sun. The magnitude of the vector is the distance between the two points and the direction refers to the direction of displacement from A to B. Many algebraic operations on real numbers such as addition, subtraction, multiplication, and negation have close analogues for vectors, operations which obey the familiar algebraic laws of commutativity, associativity, and distributivity. These operations and associated laws qualify Euclidean vectors as an example of the more generalized concept of vectors defined simply as elements of a vector space.

The algebraically imaginary part, being geometrically constructed by a straight line, or radius vector, which has, in general, for each determined quaternion, a determined length and determined direction in space, may be called the vector part, or simply the vector of the quaternion.
(1, 2, 3) + (−2, 0, 4) = (1 − 2, 2 + 0, 3 + 4) = (−1, 2, 7).
${\displaystyle \mathbf {a} =(2,3).}$
${\displaystyle \mathbf {a} =(a_{1},a_{2},a_{3}).}$
also written
${\displaystyle \mathbf {a} =(a_{\text{x}},a_{\text{y}},a_{\text{z}}).}$
${\displaystyle \mathbf {a} =(a_{1},a_{2},a_{3},\cdots ,a_{n-1},a_{n}).}$
${\displaystyle \mathbf {a} ={\begin{bmatrix}a_{1}\\a_{2}\\a_{3}\\\end{bmatrix}}=[a_{1}\ a_{2}\ a_{3}].}$
${\displaystyle {\mathbf {e} }_{1}=(1,0,0),\ {\mathbf {e} }_{2}=(0,1,0),\ {\mathbf {e} }_{3}=(0,0,1).}$
${\displaystyle \mathbf {a} =(a_{1},a_{2},a_{3})=a_{1}(1,0,0)+a_{2}(0,1,0)+a_{3}(0,0,1),\ }$
${\displaystyle \mathbf {a} =\mathbf {a} _{1}+\mathbf {a} _{2}+\mathbf {a} _{3}=a_{1}{\mathbf {e} }_{1}+a_{2}{\mathbf {e} }_{2}+a_{3}{\mathbf {e} }_{3},}$
${\displaystyle \mathbf {a} =\mathbf {a} _{\text{x}}+\mathbf {a} _{\text{y}}+\mathbf {a} _{\text{z}}=a_{\text{x}}{\mathbf {i} }+a_{\text{y}}{\mathbf {j} }+a_{\text{z}}{\mathbf {k} }.}$
${\displaystyle {\mathbf {e} }_{1}=(1,0,0),\ {\mathbf {e} }_{2}=(0,1,0),\ {\mathbf {e} }_{3}=(0,0,1)}$
${\displaystyle {\mathbf {a} }=a_{1}{\mathbf {e} }_{1}+a_{2}{\mathbf {e} }_{2}+a_{3}{\mathbf {e} }_{3}.}$
${\displaystyle {\mathbf {a} }=a_{1}{\mathbf {e} }_{1}+a_{2}{\mathbf {e} }_{2}+a_{3}{\mathbf {e} }_{3}}$
${\displaystyle {\mathbf {b} }=b_{1}{\mathbf {e} }_{1}+b_{2}{\mathbf {e} }_{2}+b_{3}{\mathbf {e} }_{3}}$
${\displaystyle a_{1}=b_{1},\quad a_{2}=b_{2},\quad a_{3}=b_{3}.\,}$
${\displaystyle {\mathbf {a} }=a_{1}{\mathbf {e} }_{1}+a_{2}{\mathbf {e} }_{2}+a_{3}{\mathbf {e} }_{3}}$
${\displaystyle {\mathbf {b} }=b_{1}{\mathbf {e} }_{1}+b_{2}{\mathbf {e} }_{2}+b_{3}{\mathbf {e} }_{3}}$
${\displaystyle a_{1}=-b_{1},\quad a_{2}=-b_{2},\quad a_{3}=-b_{3}.\,}$
${\displaystyle \left\|\mathbf {a} \right\|={\sqrt {{a_{1}}^{2}+{a_{2}}^{2}+{a_{3}}^{2}}}}$
${\displaystyle \left\|\mathbf {a} \right\|={\sqrt {\mathbf {a} \cdot \mathbf {a} }}.}$
Unit vector
${\displaystyle \mathbf {\hat {a}} ={\frac {\mathbf {a} }{\left\|\mathbf {a} \right\|}}={\frac {a_{1}}{\left\|\mathbf {a} \right\|}}\mathbf {e} _{1}+{\frac {a_{2}}{\left\|\mathbf {a} \right\|}}\mathbf {e} _{2}+{\frac {a_{3}}{\left\|\mathbf {a} \right\|}}\mathbf {e} _{3}}$
Zero vector
${\displaystyle \mathbf {a} +\mathbf {b} =(a_{1}+b_{1})\mathbf {e} _{1}+(a_{2}+b_{2})\mathbf {e} _{2}+(a_{3}+b_{3})\mathbf {e} _{3}.}$
${\displaystyle \mathbf {a} -\mathbf {b} =(a_{1}-b_{1})\mathbf {e} _{1}+(a_{2}-b_{2})\mathbf {e} _{2}+(a_{3}-b_{3})\mathbf {e} _{3}.}$
${\displaystyle r\mathbf {a} =(ra_{1})\mathbf {e} _{1}+(ra_{2})\mathbf {e} _{2}+(ra_{3})\mathbf {e} _{3}.}$
${\displaystyle \mathbf {a} \cdot \mathbf {b} =\left\|\mathbf {a} \right\|\left\|\mathbf {b} \right\|\cos \theta }$
${\displaystyle \mathbf {a} \cdot \mathbf {b} =a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3}.}$
${\displaystyle \mathbf {a} \times \mathbf {b} =\left\|\mathbf {a} \right\|\left\|\mathbf {b} \right\|\sin(\theta )\,\mathbf {n} }$
${\displaystyle {\mathbf {a} }\times {\mathbf {b} }=(a_{2}b_{3}-a_{3}b_{2}){\mathbf {e} }_{1}+(a_{3}b_{1}-a_{1}b_{3}){\mathbf {e} }_{2}+(a_{1}b_{2}-a_{2}b_{1}){\mathbf {e} }_{3}.}$
${\displaystyle \mathbf {a} =p\mathbf {e} _{1}+q\mathbf {e} _{2}+r\mathbf {e} _{3}}$.
${\displaystyle p=\mathbf {a} \cdot \mathbf {e} _{1}}$,
${\displaystyle q=\mathbf {a} \cdot \mathbf {e} _{2}}$,
${\displaystyle r=\mathbf {a} \cdot \mathbf {e} _{3}}$.
${\displaystyle \mathbf {a} =u\mathbf {n} _{1}+v\mathbf {n} _{2}+w\mathbf {n} _{3}}$
${\displaystyle u=\mathbf {a} \cdot \mathbf {n} _{1}}$,
${\displaystyle v=\mathbf {a} \cdot \mathbf {n} _{2}}$,
${\displaystyle w=\mathbf {a} \cdot \mathbf {n} _{3}}$.
${\displaystyle u=(p\mathbf {e} _{1}+q\mathbf {e} _{2}+r\mathbf {e} _{3})\cdot \mathbf {n} _{1}}$,
${\displaystyle v=(p\mathbf {e} _{1}+q\mathbf {e} _{2}+r\mathbf {e} _{3})\cdot \mathbf {n} _{2}}$,
${\displaystyle w=(p\mathbf {e} _{1}+q\mathbf {e} _{2}+r\mathbf {e} _{3})\cdot \mathbf {n} _{3}}$.
${\displaystyle u=p\mathbf {e} _{1}\cdot \mathbf {n} _{1}+q\mathbf {e} _{2}\cdot \mathbf {n} _{1}+r\mathbf {e} _{3}\cdot \mathbf {n} _{1}}$,
${\displaystyle v=p\mathbf {e} _{1}\cdot \mathbf {n} _{2}+q\mathbf {e} _{2}\cdot \mathbf {n} _{2}+r\mathbf {e} _{3}\cdot \mathbf {n} _{2}}$,
${\displaystyle w=p\mathbf {e} _{1}\cdot \mathbf {n} _{3}+q\mathbf {e} _{2}\cdot \mathbf {n} _{3}+r\mathbf {e} _{3}\cdot \mathbf {n} _{3}}$.
${\displaystyle u=c_{11}p+c_{12}q+c_{13}r}$,
${\displaystyle v=c_{21}p+c_{22}q+c_{23}r}$,
${\displaystyle w=c_{31}p+c_{32}q+c_{33}r}$,
${\displaystyle {\begin{bmatrix}u\\v\\w\\\end{bmatrix}}={\begin{bmatrix}c_{11}&c_{12}&c_{13}\\c_{21}&c_{22}&c_{23}\\c_{31}&c_{32}&c_{33}\end{bmatrix}}{\begin{bmatrix}p\\q\\r\end{bmatrix}}}$.
${\displaystyle c_{11}=\mathbf {n} _{1}\cdot \mathbf {e} _{1}}$
${\displaystyle c_{12}=\mathbf {n} _{1}\cdot \mathbf {e} _{2}}$
${\displaystyle c_{13}=\mathbf {n} _{1}\cdot \mathbf {e} _{3}}$
${\displaystyle c_{21}=\mathbf {n} _{2}\cdot \mathbf {e} _{1}}$
${\displaystyle c_{22}=\mathbf {n} _{2}\cdot \mathbf {e} _{2}}$
${\displaystyle c_{23}=\mathbf {n} _{2}\cdot \mathbf {e} _{3}}$
${\displaystyle c_{31}=\mathbf {n} _{3}\cdot \mathbf {e} _{1}}$
${\displaystyle c_{32}=\mathbf {n} _{3}\cdot \mathbf {e} _{2}}$
${\displaystyle c_{33}=\mathbf {n} _{3}\cdot \mathbf {e} _{3}}$
${\displaystyle {\mathbf {x} }=x_{1}{\mathbf {e} }_{1}+x_{2}{\mathbf {e} }_{2}+x_{3}{\mathbf {e} }_{3}.}$
${\displaystyle {\mathbf {y} }-{\mathbf {x} }=(y_{1}-x_{1}){\mathbf {e} }_{1}+(y_{2}-x_{2}){\mathbf {e} }_{2}+(y_{3}-x_{3}){\mathbf {e} }_{3}.}$
${\displaystyle {\mathbf {x} }_{t}=t{\mathbf {v} }+{\mathbf {x} }_{0},}$
${\displaystyle {\mathbf {F} }=m{\mathbf {a} }}$
${\displaystyle E={\mathbf {F} }\cdot ({\mathbf {x} }_{2}-{\mathbf {x} }_{1}).}$
${\displaystyle {\frac {df}{d\tau }}=\sum _{\alpha =1}^{n}{\frac {dx^{\alpha }}{d\tau }}{\frac {\partial f}{\partial x^{\alpha }}}.}$
${\displaystyle t^{\alpha }={\frac {dx^{\alpha }}{d\tau }}.}$
${\displaystyle {\frac {d}{d\tau }}=\sum _{\alpha }t^{\alpha }{\frac {\partial }{\partial x^{\alpha }}}.}$
${\displaystyle \mathbf {a} \equiv a^{\alpha }{\frac {\partial }{\partial x^{\alpha }}}.}$
Wikipedia