Sivu:Lukion taulukot.pdf/31

30 MERKINTÖJÄ, MÄÄRITELMIÄ JA KAAVOJA

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Vektorilaskenta

Olkoon ${\bar {a}}=a_{x}{\bar {i}}+a_{y}{\bar {j}}+a_{z}{\bar {k}},\ {\bar {b}}=b_{x}{\bar {i}}+b_{y}{\bar {j}}+b_{z}{\bar {k}}{\text{ ja }}{\bar {c}}=c_{x}{\bar {i}}+c_{y}{\bar {j}}+c_{z}{\bar {k}}$

1.	$|{\bar {a}}|={\sqrt {a_{x}^{2}+a_{y}^{2}+a_{z}^{2}}}$	${\bar {a}}$:n pituus
2.	${\bar {a}}^{\circ }={\frac {\bar {a}}{|{\bar {a}}|}}$	${\bar {a}}$:n suuntainen yksikkövektori
3.	${\begin{aligned}{\bar {a}}={\bar {b}}&\Leftrightarrow {\bar {a}}\upuparrows {\bar {b}}\land |{\bar {a}}|=|{\bar {b}}|\\&\Leftrightarrow (a_{x}=b_{x}\land a_{y}=b_{y}\land a_{z}=b_{z})\end{aligned}}$	${\bar {a}}$ ja ${\bar {b}}$ identtiset
4.	${\bar {b}}=r{\bar {a}}\quad ({\bar {a}},{\bar {b}}\neq {\bar {0}},r\neq 0)$	yhdensuuntaisuusehto
5.	${\bar {a}}+{\bar {b}}=(a_{x}+b_{x}){\bar {i}}+(a_{y}+b_{y}){\bar {j}}+(a_{z}+b_{z}){\bar {k}}$	summavektori
6.	$r{\bar {a}}=ra_{x}{\bar {i}}+ra_{y}{\bar {j}}+ra_{z}{\bar {k}}$	vektorin ${\bar {a}}$ kertominen luvulla $r$
7.	${\bar {a}}\cdot {\bar {b}}=|{\bar {a}}||{\bar {b}}|\cos({\bar {a}},{\bar {b}})=a_{x}b_{x}+a_{y}b_{y}+a_{z}b_{z}$	piste- eli skalaaritulo
	${\bar {a}}\cdot {\bar {b}}=0\Leftrightarrow {\bar {a}}\perp {\bar {b}},({\bar {a}},{\bar {b}}\neq {\bar {0}})$	kohtisuoruusehto
	$\cos({\bar {a}},{\bar {b}})={\frac {{\bar {a}}\cdot {\bar {b}}}{|{\bar {a}}||{\bar {b}}|}}\;(0^{\circ }\leq ({\bar {a}},{\bar {b}})\leq 180^{\circ })$	${\bar {a}}$:n ja ${\bar {b}}$:n välisen kulman kosini
	$a_{b}={\frac {{\bar {a}}\cdot {\bar {b}}}{|{\bar {b}}|}}$	${\bar {a}}$:n skalaariprojektio ${\bar {b}}$:llä
	${\bar {a}}_{b}={\frac {{\bar {a}}\cdot {\bar {b}}}{{\bar {b}}\cdot {\bar {b}}}}{\bar {b}}$	${\bar {a}}$:n vektoriprojektio ${\bar {b}}$:llä
8.	${\begin{aligned}{\bar {a}}\times {\bar {b}}&=|{\bar {a}}||{\bar {b}}|\sin({\bar {a}},{\bar {b}}){\bar {e}}={\begin{vmatrix}{\bar {i}}&{\bar {j}}&{\bar {k}}\\a_{x}&a_{y}&a_{z}\\b_{x}&b_{y}&b_{z}\end{vmatrix}}\\&={\begin{vmatrix}a_{y}&a_{z}\\b_{y}&b_{z}\end{vmatrix}}{\bar {i}}+{\begin{vmatrix}a_{z}&a_{x}\\b_{z}&b_{x}\end{vmatrix}}{\bar {j}}+{\begin{vmatrix}a_{x}&a_{y}\\b_{x}&b_{y}\end{vmatrix}}{\bar {k}}\end{aligned}}$	risti- eli vektoritulo
	${\bar {a}}\times {\bar {b}}={\bar {0}}\Leftrightarrow {\bar {a}}\parallel {\bar {b}}\;({\bar {a}},{\bar {b}}\neq {\bar {0}})$	yhdensuuntaisuusehto
	$A=|{\bar {a}}\times {\bar {b}}|$	$A$ on ${\bar {a}}$:n ja ${\bar {b}}$:n määräämän suunnikkaan ala