Utilisateur:RichardHoule

Propriété des vecteurs

$\|k{\overrightarrow {v}}\|=|k|\cdot \|{\overrightarrow {v}}\|$
${\overrightarrow {u}}+{\overrightarrow {v}}={\overrightarrow {v}}+{\overrightarrow {u}}$
${\overrightarrow {u}}+({\overrightarrow {v}}+{\overrightarrow {w}})=({\overrightarrow {u}}+{\overrightarrow {v}})+{\overrightarrow {w}}$
${\overrightarrow {u}}+{\overrightarrow {0}}={\overrightarrow {u}}$
${\overrightarrow {u}}-{\overrightarrow {u}}={\overrightarrow {0}}$
$k({\overrightarrow {u}}+{\overrightarrow {v}})=k{\overrightarrow {u}}+k{\overrightarrow {v}}$
$(k+r){\overrightarrow {u}}=k{\overrightarrow {u}}+k{\overrightarrow {u}}$
$rs{\overrightarrow {u}}=r(s{\overrightarrow {u}})$

Géométrie

${\overrightarrow {AB}}={\overrightarrow {OB}}-{\overrightarrow {OA}}$

Barycentre

${\overrightarrow {OP}}={\dfrac {1}{n}}({\overrightarrow {OP_{1}}}+{\overrightarrow {OP_{2}}}+\ldots +{\overrightarrow {OP_{n}}})$

Vecteurs

Norme

$\|{\overrightarrow {u}}\|={\sqrt {(a_{1})^{2}+(a_{2})^{2}+\dots +(a_{n})^{2}}}$
$\|{\overrightarrow {u}}\|=\|{\overrightarrow {u}}_{\perp }\|$
${\frac {\overrightarrow {u}}{\|{\overrightarrow {u}}\|}}=1$

Produit scalaire

${\overrightarrow {u}}\cdot {\overrightarrow {v}}=\|{\overrightarrow {u}}\|\cdot \|{\overrightarrow {v}}\|\cos \theta$
${\overrightarrow {u}}\cdot {\overrightarrow {v}}=u_{1}v_{1}+u_{2}v_{2}+\ldots +u_{n}v_{n}$
${\overrightarrow {u}}\cdot {\overrightarrow {v}}={\overrightarrow {v}}\cdot {\overrightarrow {u}}$
$(k{\overrightarrow {u}})\cdot {\overrightarrow {v}}={\overrightarrow {u}}\cdot (k{\overrightarrow {v}})=k({\overrightarrow {u}}\cdot {\overrightarrow {v}})$
${\overrightarrow {u}}\cdot {\overrightarrow {u}}=\|{\overrightarrow {u}}^{2}\|$

Angle entre deux vecteurs

$\theta =\arccos({\frac {{\overrightarrow {u}}\cdot {\overrightarrow {v}}}{\|{\overrightarrow {u}}\|\cdot \|{\overrightarrow {v}}\|}})$

Loi des cosinus

$c^{2}=a^{2}+b^{2}-2ab\cos \theta$

Perpendiculaire dans $\mathbb {R} ^{2}$ (Orthogonaux)

${\overrightarrow {u}}=(a,b)\;\;{\overrightarrow {u}}_{\perp }=(-b,a)$

Projection orthogonale

${\overrightarrow {v}}_{\overrightarrow {u}}\parallel {\overrightarrow {u}}$

${\overrightarrow {v}}_{\overrightarrow {u}}=({\frac {{\overrightarrow {u}}\cdot {\overrightarrow {v}}}{\|{\overrightarrow {u}}\|^{2}}}){\overrightarrow {u}}$

$\|{\overrightarrow {v}}_{\overrightarrow {u}}\|={\frac {|{\overrightarrow {u}}\cdot {\overrightarrow {v}}|}{\|{\overrightarrow {u}}\|}}$

Produit vectoriel

Aire d'un parallélogramme dans $\mathbb {R} _{3}$

$\|{\overrightarrow {u}}\wedge {\overrightarrow {v}}\|=\|{\overrightarrow {u}}\|\cdot \|{\overrightarrow {v}}\|\sin \theta$
${\overrightarrow {u}}\wedge {\overrightarrow {u}}=0$
${\overrightarrow {u}}\wedge {\overrightarrow {v}}=-({\overrightarrow {v}}\wedge {\overrightarrow {u}})$
$k{\overrightarrow {u}}\wedge {\overrightarrow {v}}={\overrightarrow {u}}\wedge k{\overrightarrow {v}}=k({\overrightarrow {u}}\wedge {\overrightarrow {v}})$
${\overrightarrow {u}}\wedge ({\overrightarrow {v}}+{\overrightarrow {w}})=({\overrightarrow {u}}\wedge {\overrightarrow {v}})+({\overrightarrow {u}}\wedge {\overrightarrow {w}})$
${\overrightarrow {i}}\wedge {\overrightarrow {j}}={\overrightarrow {k}},{\overrightarrow {i}}\wedge {\overrightarrow {k}}=-{\overrightarrow {j}},{\overrightarrow {j}}\wedge {\overrightarrow {k}}={\overrightarrow {i}}$
$(u_{1},u_{2},u_{3})\wedge (v_{1},v_{2},v_{3})={\begin{vmatrix}{\overrightarrow {i}}&{\overrightarrow {j}}&{\overrightarrow {k}}\\u_{1}&u_{2}&u_{3}\\v_{1}&v_{2}&v_{3}\\\end{vmatrix}}=({\begin{vmatrix}u_{2}&u_{3}\\v_{2}&v_{3}\end{vmatrix}},-{\begin{vmatrix}u_{1}&u_{3}\\v_{1}&v_{3}\end{vmatrix}},{\begin{vmatrix}u_{1}&u_{2}\\v_{1}&v_{2}\end{vmatrix}})$

Si le produit vectoriel est égal à zéro, alors les deux vecteurs sont parallèles.

Produit mixte

$({\overrightarrow {u}}\wedge {\overrightarrow {v}})\cdot {\overrightarrow {w}}=\|{\overrightarrow {u}}\wedge {\overrightarrow {v}}\|\cdot \|{\overrightarrow {w}}\|\cos \alpha$
$({\overrightarrow {u}}\wedge {\overrightarrow {v}})\cdot {\overrightarrow {w}}=\Delta \langle {\overrightarrow {u}},{\overrightarrow {v}},{\overrightarrow {w}}\rangle ={\begin{vmatrix}u_{1}&u_{2}&u_{3}\\v_{1}&v_{2}&v_{3}\\w_{1}&w_{2}&w_{3}\\\end{vmatrix}}$

Matrices

Déterminant 2x2

$\Delta \langle {\overrightarrow {u}},{\overrightarrow {v}}\rangle ={\begin{vmatrix}{\overrightarrow {u}}_{1}&{\overrightarrow {u}}_{2}\\{\overrightarrow {v}}_{1}&{\overrightarrow {v}}_{2}\end{vmatrix}}$

$\Delta \langle {\overrightarrow {u}},{\overrightarrow {u}}\rangle ={\begin{vmatrix}a&b\\a&b\end{vmatrix}}=0$

$\Delta \langle {\overrightarrow {ku}},{\overrightarrow {v}}\rangle =\Delta \langle {\overrightarrow {u}},{\overrightarrow {kv}}\rangle =k\Delta \langle {\overrightarrow {u}},{\overrightarrow {v}}\rangle$

$\Delta \langle {\overrightarrow {u}},{\overrightarrow {v}}\rangle =-\Delta \langle {\overrightarrow {v}},{\overrightarrow {u}}\rangle$

$\Delta \langle {\overrightarrow {u}},{\overrightarrow {v}}\rangle =\Delta \langle {\overrightarrow {u}}+k{\overrightarrow {v}},{\overrightarrow {v}}\rangle =\Delta \langle {\overrightarrow {u}},{\overrightarrow {v}}+k{\overrightarrow {u}}\rangle$

${\begin{vmatrix}ka&kb\\c&d\end{vmatrix}}={\begin{vmatrix}a&b\\kc&kd\end{vmatrix}}=k{\begin{vmatrix}a&b\\c&d\end{vmatrix}}$

${\begin{vmatrix}r_{1}+s_{1}&r_{2}+s_{2}\\c&d\end{vmatrix}}={\begin{vmatrix}r_{1}&r_{2}\\c&d\end{vmatrix}}+{\begin{vmatrix}s_{1}&s_{2}\\c&d\end{vmatrix}}$

${\begin{vmatrix}a&b\\c&d\end{vmatrix}}={\begin{vmatrix}a+kc&b+kd\\c&d\end{vmatrix}}={\begin{vmatrix}a&b\\c+ka&d+kb\end{vmatrix}}$

${\begin{vmatrix}a&b\\c&d\end{vmatrix}}=ad-bc$

${\begin{vmatrix}a&b\\c&d\end{vmatrix}}={\begin{vmatrix}a&c\\b&d\end{vmatrix}}$

Déterminant 3x3

$\Delta \langle {\overrightarrow {u}},{\overrightarrow {u}},{\overrightarrow {v}}\rangle =\Delta \langle {\overrightarrow {u}},{\overrightarrow {v}},{\overrightarrow {u}}\rangle =\Delta \langle {\overrightarrow {u}},{\overrightarrow {v}},{\overrightarrow {v}}\rangle =0$
${\begin{vmatrix}a&b&c\\a&b&c\\e&f&g\end{vmatrix}}={\begin{vmatrix}a&b&c\\e&f&g\\a&b&c\end{vmatrix}}={\begin{vmatrix}a&b&c\\e&f&g\\e&f&g\end{vmatrix}}=0$

$\Delta \langle k{\overrightarrow {u}},{\overrightarrow {v}},{\overrightarrow {w}}\rangle =\Delta \langle {\overrightarrow {u}},k{\overrightarrow {v}},{\overrightarrow {w}}\rangle =\Delta \langle {\overrightarrow {u}},{\overrightarrow {v}},k{\overrightarrow {w}}\rangle =k\Delta \langle {\overrightarrow {u}},{\overrightarrow {v}},{\overrightarrow {w}}\rangle$
${\begin{vmatrix}ka&kb&kc\\d&e&f\\g&h&i\end{vmatrix}}={\begin{vmatrix}a&b&c\\kd&ke&kf\\g&h&i\end{vmatrix}}={\begin{vmatrix}a&b&c\\d&e&f\\kg&kh&ki\end{vmatrix}}=k{\begin{vmatrix}a&b&c\\d&e&f\\g&h&i\end{vmatrix}}$

$\Delta \langle {\overrightarrow {r}}+{\overrightarrow {s}},{\overrightarrow {v}},{\overrightarrow {w}}\rangle =\Delta \langle {\overrightarrow {r}},{\overrightarrow {v}},{\overrightarrow {w}}\rangle +\Delta \langle {\overrightarrow {s}},{\overrightarrow {v}},{\overrightarrow {w}}\rangle$
${\begin{vmatrix}r_{1}+s_{1}&r_{2}+s_{2}&r_{3}+s_{3}\\v_{1}&v_{2}&v_{3}\\w_{1}&w_{2}&w_{3}\end{vmatrix}}={\begin{vmatrix}r_{1}&r_{2}&r_{3}\\v_{1}&v_{2}&v_{3}\\w_{1}&w_{2}&w_{3}\end{vmatrix}}+{\begin{vmatrix}s_{1}&s_{2}&s_{3}\\v_{1}&v_{2}&v_{3}\\w_{1}&w_{2}&w_{3}\end{vmatrix}}$

$\Delta \langle {\overrightarrow {u}},{\overrightarrow {v}},{\overrightarrow {w}}\rangle =-\Delta \langle {\overrightarrow {v}},{\overrightarrow {u}},{\overrightarrow {w}}\rangle =-\Delta \langle {\overrightarrow {w}},{\overrightarrow {v}},{\overrightarrow {u}}\rangle =-\Delta \langle {\overrightarrow {u}},{\overrightarrow {w}},{\overrightarrow {v}}\rangle$
${\begin{vmatrix}u_{1}&u_{2}&u_{3}\\v_{1}&v_{2}&v_{3}\\w_{1}&w_{2}&w_{3}\end{vmatrix}}=-{\begin{vmatrix}v_{1}&v_{2}&v_{3}\\u_{1}&u_{2}&u_{3}\\w_{1}&w_{2}&w_{3}\end{vmatrix}}=-{\begin{vmatrix}w_{1}&w_{2}&w_{3}\\v_{1}&v_{2}&v_{3}\\u_{1}&u_{2}&u_{3}\end{vmatrix}}=-{\begin{vmatrix}u_{1}&u_{2}&u_{3}\\w_{1}&w_{2}&w_{3}\\v_{1}&v_{2}&v_{3}\end{vmatrix}}$

$\Delta \langle {\overrightarrow {u}},{\overrightarrow {v}},{\overrightarrow {w}}\rangle =\Delta \langle {\overrightarrow {u}}+a{\overrightarrow {v}}+b{\overrightarrow {w}},{\overrightarrow {v}},{\overrightarrow {w}}\rangle$
${\begin{vmatrix}u_{1}&u_{2}&u_{3}\\v_{1}&v_{2}&v_{3}\\w_{1}&w_{2}&w_{3}\end{vmatrix}}=-{\begin{vmatrix}u_{1}+av_{1}+bw_{2}&u_{2}+av_{2}+bw_{2}&u_{3}+av_{3}+bw_{3}\\v_{1}&v_{2}&v_{3}\\w_{1}&w_{2}&w_{3}\end{vmatrix}}$

${\begin{vmatrix}u_{1}&u_{2}&u_{3}\\v_{1}&v_{2}&v_{3}\\w_{1}&w_{2}&w_{3}\end{vmatrix}}=u_{1}{\begin{vmatrix}v_{2}&v_{3}\\w_{2}&w_{3}\end{vmatrix}}-u_{2}{\begin{vmatrix}v_{1}&v_{3}\\w_{1}&w_{3}\end{vmatrix}}+u_{3}{\begin{vmatrix}v_{1}&v_{2}\\w_{1}&w_{2}\end{vmatrix}}$

Règle de Cramer

Système de deux équations à deux variables

$ax+by=m$
$cx+dy=n$

$x={\frac {\begin{vmatrix}m&b\\n&d\end{vmatrix}}{\begin{vmatrix}a&b\\c&d\end{vmatrix}}}\;\;y={\frac {\begin{vmatrix}a&m\\c&n\end{vmatrix}}{\begin{vmatrix}a&b\\c&d\end{vmatrix}}}$

Système de trois équations à trois variables

$ax+by+cz=m$
$dx+ey+fz=n$
$gx+hy+iz=p$

$x={\frac {\begin{vmatrix}m&b&c\\n&e&f\\p&h&i\end{vmatrix}}{\begin{vmatrix}a&b&c\\d&e&f\\g&h&i\end{vmatrix}}}\;\;y={\frac {\begin{vmatrix}a&m&c\\d&n&f\\g&p&i\end{vmatrix}}{\begin{vmatrix}a&b&c\\d&e&f\\g&h&i\end{vmatrix}}}\;\;z={\frac {\begin{vmatrix}a&b&m\\d&e&n\\g&h&p\end{vmatrix}}{\begin{vmatrix}a&b&c\\d&e&f\\g&h&i\end{vmatrix}}}$

Équation vectorielles

$(x,y)=(x_{0},y_{0})+t(a,b)$
$(x,y,z)=(x_{0},y_{0},z_{0})+t(a,b,c)$
$y=mx+b\Rightarrow (x,y)=(0,b)+t(1,m)$

Équation symétrique

$\mathbb {R} ^{2}:{\frac {x-x_{0}}{a}}={\frac {y-y_{0}}{b}}$

$\mathbb {R} ^{3}:{\frac {x-x_{0}}{a}}={\frac {y-y_{0}}{b}}={\frac {z-z_{0}}{c}}$

Équation normale

$ax+by+c=0\;o{\grave {u}}\;c=-ax_{0}-by_{0}$

Distance d'un point à partir d'une droite

$\mathbb {R} _{2}:d(P,{\mathcal {D}})={\frac {|\Delta \langle {\overrightarrow {AP}},{\overrightarrow {v}}\rangle |}{\|{\overrightarrow {v}}\|}}$
$\mathbb {R} _{3}:d(P,{\mathcal {D}})={\frac {\|{\overrightarrow {AP}}\wedge {\overrightarrow {v}}\|}{\|{\overrightarrow {v}}\|}}$

Distance entre deux droites

$d({\mathcal {D}}_{1},{\mathcal {D}}_{2})=\|{\overrightarrow {P_{1}P_{2}}}\|={\frac {|{\overrightarrow {A_{1}A_{2}}}\cdot ({\overrightarrow {v_{1}}}\wedge {\overrightarrow {v_{2}}})|}{\|{\overrightarrow {v_{1}}}\wedge {\overrightarrow {v_{2}}}\|}}$