‖ k v → ‖ = | k | ⋅ ‖ v → ‖ {\displaystyle \|k{\overrightarrow {v}}\|=|k|\cdot \|{\overrightarrow {v}}\|} u → + v → = v → + u → {\displaystyle {\overrightarrow {u}}+{\overrightarrow {v}}={\overrightarrow {v}}+{\overrightarrow {u}}} u → + ( v → + w → ) = ( u → + v → ) + w → {\displaystyle {\overrightarrow {u}}+({\overrightarrow {v}}+{\overrightarrow {w}})=({\overrightarrow {u}}+{\overrightarrow {v}})+{\overrightarrow {w}}} u → + 0 → = u → {\displaystyle {\overrightarrow {u}}+{\overrightarrow {0}}={\overrightarrow {u}}} u → − u → = 0 → {\displaystyle {\overrightarrow {u}}-{\overrightarrow {u}}={\overrightarrow {0}}} k ( u → + v → ) = k u → + k v → {\displaystyle k({\overrightarrow {u}}+{\overrightarrow {v}})=k{\overrightarrow {u}}+k{\overrightarrow {v}}} ( k + r ) u → = k u → + k u → {\displaystyle (k+r){\overrightarrow {u}}=k{\overrightarrow {u}}+k{\overrightarrow {u}}} r s u → = r ( s u → ) {\displaystyle rs{\overrightarrow {u}}=r(s{\overrightarrow {u}})}
A B → = O B → − O A → {\displaystyle {\overrightarrow {AB}}={\overrightarrow {OB}}-{\overrightarrow {OA}}}
O P → = 1 n ( O P 1 → + O P 2 → + … + O P n → ) {\displaystyle {\overrightarrow {OP}}={\dfrac {1}{n}}({\overrightarrow {OP_{1}}}+{\overrightarrow {OP_{2}}}+\ldots +{\overrightarrow {OP_{n}}})}
‖ u → ‖ = ( a 1 ) 2 + ( a 2 ) 2 + ⋯ + ( a n ) 2 {\displaystyle \|{\overrightarrow {u}}\|={\sqrt {(a_{1})^{2}+(a_{2})^{2}+\dots +(a_{n})^{2}}}} ‖ u → ‖ = ‖ u → ⊥ ‖ {\displaystyle \|{\overrightarrow {u}}\|=\|{\overrightarrow {u}}_{\perp }\|} u → ‖ u → ‖ = 1 {\displaystyle {\frac {\overrightarrow {u}}{\|{\overrightarrow {u}}\|}}=1}
u → ⋅ v → = ‖ u → ‖ ⋅ ‖ v → ‖ cos θ {\displaystyle {\overrightarrow {u}}\cdot {\overrightarrow {v}}=\|{\overrightarrow {u}}\|\cdot \|{\overrightarrow {v}}\|\cos \theta } u → ⋅ v → = u 1 v 1 + u 2 v 2 + … + u n v n {\displaystyle {\overrightarrow {u}}\cdot {\overrightarrow {v}}=u_{1}v_{1}+u_{2}v_{2}+\ldots +u_{n}v_{n}} u → ⋅ v → = v → ⋅ u → {\displaystyle {\overrightarrow {u}}\cdot {\overrightarrow {v}}={\overrightarrow {v}}\cdot {\overrightarrow {u}}} ( k u → ) ⋅ v → = u → ⋅ ( k v → ) = k ( u → ⋅ v → ) {\displaystyle (k{\overrightarrow {u}})\cdot {\overrightarrow {v}}={\overrightarrow {u}}\cdot (k{\overrightarrow {v}})=k({\overrightarrow {u}}\cdot {\overrightarrow {v}})} u → ⋅ u → = ‖ u → 2 ‖ {\displaystyle {\overrightarrow {u}}\cdot {\overrightarrow {u}}=\|{\overrightarrow {u}}^{2}\|}
θ = arccos ( u → ⋅ v → ‖ u → ‖ ⋅ ‖ v → ‖ ) {\displaystyle \theta =\arccos({\frac {{\overrightarrow {u}}\cdot {\overrightarrow {v}}}{\|{\overrightarrow {u}}\|\cdot \|{\overrightarrow {v}}\|}})}
c 2 = a 2 + b 2 − 2 a b cos θ {\displaystyle c^{2}=a^{2}+b^{2}-2ab\cos \theta }
u → = ( a , b ) u → ⊥ = ( − b , a ) {\displaystyle {\overrightarrow {u}}=(a,b)\;\;{\overrightarrow {u}}_{\perp }=(-b,a)}
v → u → ∥ u → {\displaystyle {\overrightarrow {v}}_{\overrightarrow {u}}\parallel {\overrightarrow {u}}} v → u → = ( u → ⋅ v → ‖ u → ‖ 2 ) u → {\displaystyle {\overrightarrow {v}}_{\overrightarrow {u}}=({\frac {{\overrightarrow {u}}\cdot {\overrightarrow {v}}}{\|{\overrightarrow {u}}\|^{2}}}){\overrightarrow {u}}} ‖ v → u → ‖ = | u → ⋅ v → | ‖ u → ‖ {\displaystyle \|{\overrightarrow {v}}_{\overrightarrow {u}}\|={\frac {|{\overrightarrow {u}}\cdot {\overrightarrow {v}}|}{\|{\overrightarrow {u}}\|}}}
Aire d'un parallélogramme dans R 3 {\displaystyle \mathbb {R} _{3}}
‖ u → ∧ v → ‖ = ‖ u → ‖ ⋅ ‖ v → ‖ sin θ {\displaystyle \|{\overrightarrow {u}}\wedge {\overrightarrow {v}}\|=\|{\overrightarrow {u}}\|\cdot \|{\overrightarrow {v}}\|\sin \theta } u → ∧ u → = 0 {\displaystyle {\overrightarrow {u}}\wedge {\overrightarrow {u}}=0} u → ∧ v → = − ( v → ∧ u → ) {\displaystyle {\overrightarrow {u}}\wedge {\overrightarrow {v}}=-({\overrightarrow {v}}\wedge {\overrightarrow {u}})} k u → ∧ v → = u → ∧ k v → = k ( u → ∧ v → ) {\displaystyle k{\overrightarrow {u}}\wedge {\overrightarrow {v}}={\overrightarrow {u}}\wedge k{\overrightarrow {v}}=k({\overrightarrow {u}}\wedge {\overrightarrow {v}})} u → ∧ ( v → + w → ) = ( u → ∧ v → ) + ( u → ∧ w → ) {\displaystyle {\overrightarrow {u}}\wedge ({\overrightarrow {v}}+{\overrightarrow {w}})=({\overrightarrow {u}}\wedge {\overrightarrow {v}})+({\overrightarrow {u}}\wedge {\overrightarrow {w}})} i → ∧ j → = k → , i → ∧ k → = − j → , j → ∧ k → = i → {\displaystyle {\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 ) ∧ ( v 1 , v 2 , v 3 ) = | i → j → k → u 1 u 2 u 3 v 1 v 2 v 3 | = ( | u 2 u 3 v 2 v 3 | , − | u 1 u 3 v 1 v 3 | , | u 1 u 2 v 1 v 2 | ) {\displaystyle (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.
( u → ∧ v → ) ⋅ w → = ‖ u → ∧ v → ‖ ⋅ ‖ w → ‖ cos α {\displaystyle ({\overrightarrow {u}}\wedge {\overrightarrow {v}})\cdot {\overrightarrow {w}}=\|{\overrightarrow {u}}\wedge {\overrightarrow {v}}\|\cdot \|{\overrightarrow {w}}\|\cos \alpha } ( u → ∧ v → ) ⋅ w → = Δ ⟨ u → , v → , w → ⟩ = | u 1 u 2 u 3 v 1 v 2 v 3 w 1 w 2 w 3 | {\displaystyle ({\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}}}
Δ ⟨ u → , v → ⟩ = | u → 1 u → 2 v → 1 v → 2 | {\displaystyle \Delta \langle {\overrightarrow {u}},{\overrightarrow {v}}\rangle ={\begin{vmatrix}{\overrightarrow {u}}_{1}&{\overrightarrow {u}}_{2}\\{\overrightarrow {v}}_{1}&{\overrightarrow {v}}_{2}\end{vmatrix}}}
Δ ⟨ u → , u → ⟩ = | a b a b | = 0 {\displaystyle \Delta \langle {\overrightarrow {u}},{\overrightarrow {u}}\rangle ={\begin{vmatrix}a&b\\a&b\end{vmatrix}}=0}
Δ ⟨ k u → , v → ⟩ = Δ ⟨ u → , k v → ⟩ = k Δ ⟨ u → , v → ⟩ {\displaystyle \Delta \langle {\overrightarrow {ku}},{\overrightarrow {v}}\rangle =\Delta \langle {\overrightarrow {u}},{\overrightarrow {kv}}\rangle =k\Delta \langle {\overrightarrow {u}},{\overrightarrow {v}}\rangle }
Δ ⟨ u → , v → ⟩ = − Δ ⟨ v → , u → ⟩ {\displaystyle \Delta \langle {\overrightarrow {u}},{\overrightarrow {v}}\rangle =-\Delta \langle {\overrightarrow {v}},{\overrightarrow {u}}\rangle }
Δ ⟨ u → , v → ⟩ = Δ ⟨ u → + k v → , v → ⟩ = Δ ⟨ u → , v → + k u → ⟩ {\displaystyle \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 }
| k a k b c d | = | a b k c k d | = k | a b c d | {\displaystyle {\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}}}
| r 1 + s 1 r 2 + s 2 c d | = | r 1 r 2 c d | + | s 1 s 2 c d | {\displaystyle {\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}}}
| a b c d | = | a + k c b + k d c d | = | a b c + k a d + k b | {\displaystyle {\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}}}
| a b c d | = a d − b c {\displaystyle {\begin{vmatrix}a&b\\c&d\end{vmatrix}}=ad-bc}
| a b c d | = | a c b d | {\displaystyle {\begin{vmatrix}a&b\\c&d\end{vmatrix}}={\begin{vmatrix}a&c\\b&d\end{vmatrix}}}
Δ ⟨ u → , u → , v → ⟩ = Δ ⟨ u → , v → , u → ⟩ = Δ ⟨ u → , v → , v → ⟩ = 0 {\displaystyle \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} | a b c a b c e f g | = | a b c e f g a b c | = | a b c e f g e f g | = 0 {\displaystyle {\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}
Δ ⟨ k u → , v → , w → ⟩ = Δ ⟨ u → , k v → , w → ⟩ = Δ ⟨ u → , v → , k w → ⟩ = k Δ ⟨ u → , v → , w → ⟩ {\displaystyle \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 } | k a k b k c d e f g h i | = | a b c k d k e k f g h i | = | a b c d e f k g k h k i | = k | a b c d e f g h i | {\displaystyle {\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}}}
Δ ⟨ r → + s → , v → , w → ⟩ = Δ ⟨ r → , v → , w → ⟩ + Δ ⟨ s → , v → , w → ⟩ {\displaystyle \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 } | r 1 + s 1 r 2 + s 2 r 3 + s 3 v 1 v 2 v 3 w 1 w 2 w 3 | = | r 1 r 2 r 3 v 1 v 2 v 3 w 1 w 2 w 3 | + | s 1 s 2 s 3 v 1 v 2 v 3 w 1 w 2 w 3 | {\displaystyle {\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}}}
Δ ⟨ u → , v → , w → ⟩ = − Δ ⟨ v → , u → , w → ⟩ = − Δ ⟨ w → , v → , u → ⟩ = − Δ ⟨ u → , w → , v → ⟩ {\displaystyle \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 } | u 1 u 2 u 3 v 1 v 2 v 3 w 1 w 2 w 3 | = − | v 1 v 2 v 3 u 1 u 2 u 3 w 1 w 2 w 3 | = − | w 1 w 2 w 3 v 1 v 2 v 3 u 1 u 2 u 3 | = − | u 1 u 2 u 3 w 1 w 2 w 3 v 1 v 2 v 3 | {\displaystyle {\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}}}
Δ ⟨ u → , v → , w → ⟩ = Δ ⟨ u → + a v → + b w → , v → , w → ⟩ {\displaystyle \Delta \langle {\overrightarrow {u}},{\overrightarrow {v}},{\overrightarrow {w}}\rangle =\Delta \langle {\overrightarrow {u}}+a{\overrightarrow {v}}+b{\overrightarrow {w}},{\overrightarrow {v}},{\overrightarrow {w}}\rangle } | u 1 u 2 u 3 v 1 v 2 v 3 w 1 w 2 w 3 | = − | u 1 + a v 1 + b w 2 u 2 + a v 2 + b w 2 u 3 + a v 3 + b w 3 v 1 v 2 v 3 w 1 w 2 w 3 | {\displaystyle {\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}}}
| u 1 u 2 u 3 v 1 v 2 v 3 w 1 w 2 w 3 | = u 1 | v 2 v 3 w 2 w 3 | − u 2 | v 1 v 3 w 1 w 3 | + u 3 | v 1 v 2 w 1 w 2 | {\displaystyle {\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}}}
a x + b y = m {\displaystyle ax+by=m} c x + d y = n {\displaystyle cx+dy=n}
x = | m b n d | | a b c d | y = | a m c n | | a b c d | {\displaystyle 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}}}}
a x + b y + c z = m {\displaystyle ax+by+cz=m} d x + e y + f z = n {\displaystyle dx+ey+fz=n} g x + h y + i z = p {\displaystyle gx+hy+iz=p}
x = | m b c n e f p h i | | a b c d e f g h i | y = | a m c d n f g p i | | a b c d e f g h i | z = | a b m d e n g h p | | a b c d e f g h i | {\displaystyle 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}}}}
( x , y ) = ( x 0 , y 0 ) + t ( a , b ) {\displaystyle (x,y)=(x_{0},y_{0})+t(a,b)} ( x , y , z ) = ( x 0 , y 0 , z 0 ) + t ( a , b , c ) {\displaystyle (x,y,z)=(x_{0},y_{0},z_{0})+t(a,b,c)} y = m x + b ⇒ ( x , y ) = ( 0 , b ) + t ( 1 , m ) {\displaystyle y=mx+b\Rightarrow (x,y)=(0,b)+t(1,m)}
R 2 : x − x 0 a = y − y 0 b {\displaystyle \mathbb {R} ^{2}:{\frac {x-x_{0}}{a}}={\frac {y-y_{0}}{b}}} R 3 : x − x 0 a = y − y 0 b = z − z 0 c {\displaystyle \mathbb {R} ^{3}:{\frac {x-x_{0}}{a}}={\frac {y-y_{0}}{b}}={\frac {z-z_{0}}{c}}}
a x + b y + c = 0 o u ` c = − a x 0 − b y 0 {\displaystyle ax+by+c=0\;o{\grave {u}}\;c=-ax_{0}-by_{0}}
R 2 : d ( P , D ) = | Δ ⟨ A P → , v → ⟩ | ‖ v → ‖ {\displaystyle \mathbb {R} _{2}:d(P,{\mathcal {D}})={\frac {|\Delta \langle {\overrightarrow {AP}},{\overrightarrow {v}}\rangle |}{\|{\overrightarrow {v}}\|}}} R 3 : d ( P , D ) = ‖ A P → ∧ v → ‖ ‖ v → ‖ {\displaystyle \mathbb {R} _{3}:d(P,{\mathcal {D}})={\frac {\|{\overrightarrow {AP}}\wedge {\overrightarrow {v}}\|}{\|{\overrightarrow {v}}\|}}}
d ( D 1 , D 2 ) = ‖ P 1 P 2 → ‖ = | A 1 A 2 → ⋅ ( v 1 → ∧ v 2 → ) | ‖ v 1 → ∧ v 2 → ‖ {\displaystyle 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}}}\|}}}
(Orthogonaux)
