Dwie cząstki 1 i 2 zostały wysłane z jednego wspólnego położenia O i po pewnym czasie ich położenia wynoszą :
\(\displaystyle{ \vec{r}_{1} = \vec{4e}_{x} + \vec{3e}_{y} + \vec{8e}_{z}}\)
\(\displaystyle{ \vec{r}_{2} = \vec{2e}_{x} + \vec{10e}_{y} + \vec{5e}_{z}}\)
a) Narysować położenia cząstek i napisać wzór na przemieszczenie \(\displaystyle{ \vec{r} }\) cząstki 2 względem cząstki 1,
b) Znaleźć długości wektorów \(\displaystyle{ \vec{r}_{1} , \vec{r}_{2} , \vec{r} }\) stosując iloczyn skalarny,
c) Obliczyć kąty między wszystkimi możliwymi parami wektorów \(\displaystyle{ \vec{r}_{1} , \vec{r}_{2} , \vec{r} }\)
d) Obliczyć rzut wektora \(\displaystyle{ \vec{r}}\) na kierunek wektora \(\displaystyle{ \vec{r}_{1} }\)
e) Obliczyć iloczyn \(\displaystyle{ \vec{r}_{1} }\) x \(\displaystyle{ \vec{r}_{2} }\)
Wektory, iloczyny skalarne, kąty
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janusz47
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- Rejestracja: 18 mar 2009, o 16:24
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Re: Wektory, iloczyny skalarne, kąty
\(\displaystyle{ \vec{r}_{1} = 4 \vec{e}_{x} + 3\vec{e}_{y} + 8\vec{e}_{z}, }\)
\(\displaystyle{ \vec{r}_{2} = 2\vec{e}_{x} + 10\vec{e}_{y} + 5\vec{e}_{z}. }\)
a)
\(\displaystyle{ \vec{r} = \vec{r}_{2,1} = \vec{r}_{2} - \vec{r}_{1} = -2\vec{e}_{x} + 7\vec{e}_{y} - 3\vec{e}_{z}. }\)
b)
\(\displaystyle{ |\vec{r}_{1}| = \sqrt{\vec{r}_{1}\cdot \vec{r}_{1} } = \sqrt{( 4 \vec{e}_{x} + 3\vec{e}_{y} + 8\vec{e}_{z})\cdot ( 4 \vec{e}_{x} + 3\vec{e}_{y} + 8\vec{e}_{z} )} = \sqrt{4^2 + 3^2 + 8^2} = \sqrt{89}.}\)
\(\displaystyle{ |\vec{r}_{2}| = \sqrt{\vec{r}_{2}\cdot \vec{r}_{2} } = \sqrt{( 2 \vec{e}_{x} + 10\vec{e}_{y} + 5\vec{e}_{z})\cdot ( 2\vec{e}_{x} + 10\vec{e}_{y} + 5\vec{e}_{z} )} = \sqrt{2^2 + 10^2 + 5^2} = \sqrt{129}.}\)
\(\displaystyle{ |\vec{r}_{2,1}| = \sqrt{\vec{r}_{2,1}\cdot \vec{r}_{2,1} } = \sqrt{( -2 \vec{e}_{x} + 7\vec{e}_{y} - 3\vec{e}_{z})\cdot ( -2 \vec{e}_{x} + 7\vec{e}_{y} -3 \vec{e}_{z} )} = \sqrt{(-2)^2 + 7^2 + (-3)^2} = \sqrt{62}.}\)
c)
\(\displaystyle{ \cos(|\angle( \vec{r}_{1}, \vec{r}_{2})|)= \frac{\vec{r}_{1}\cdot \vec{r}_{2}}{|\vec{r}_{1}|\cdot |\vec{r}_{2}|} = \frac{(4 \vec{e}_{x} + 3\vec{e}_{y} + 8\vec{e}_{z})\cdot (2\vec{e}_{x} + 10\vec{e}_{y} + 5\vec{e}_{z})}{\sqrt{89}\cdot \sqrt{129}} = \frac{8 + 30 +40}{\sqrt{89\cdot 129}} = \frac{78}{\sqrt{11392}} \approx 0,73.}\)
\(\displaystyle{ |\angle( \vec{r}_{1}, \vec{r}_{2})|= 43^{o}. }\)
\(\displaystyle{ \cos(|\angle( \vec{r}_{1}, \vec{r}_{2,1})|)= \frac{\vec{r}_{1}\cdot \vec{r}_{2,1}}{|\vec{r}_{1}|\cdot |\vec{r}_{2,1}|} = \frac{(4 \vec{e}_{x} + 3\vec{e}_{y} + 8\vec{e}_{z})\cdot (\vec{r}_{2} - \vec{r}_{1} = -2\vec{e}_{x} + 7\vec{e}_{y} - 3\vec{e}_{z})}{\sqrt{89}\cdot \sqrt{62}} = \frac{-8 + 21 -24}{\sqrt{89\cdot 62}} = \frac{-11}{\sqrt{ 5518}} \approx- 0,15.}\)
\(\displaystyle{ |\angle( \vec{r}_{1}, \vec{r}_{2,1})|= 98,6^{o}. }\)
\(\displaystyle{ \cos(|\angle( \vec{r}_{2}, \vec{r}_{2,1})|)= \frac{\vec{r}_{2}\cdot \vec{r}_{2,1}}{|\vec{r}_{2}|\cdot |\vec{r}_{2,1}|} = \frac{( 2\vec{e}_{x} + 10\vec{e}_{y} + 5\vec{e}_{z})\cdot ( -2\vec{e}_{x} + 7\vec{e}_{y} - 3\vec{e}_{z})}{\sqrt{129}\cdot \sqrt{62}} = \frac{-4 + 70 -15}{\sqrt{129\cdot 62}} = \frac{51}{\sqrt{ 7998}} \approx 0,57 .}\)
\(\displaystyle{ |\angle( \vec{r}_{2}, \vec{r}_{2,1})|= 55,2^{o}. }\)
d)
\(\displaystyle{ rzut (\vec{r}_{2,1})_{\vec{r}_{1}} = \frac{\vec{r}_{2,1}\cdot \vec{r}_{1}}{|\vec{r}_{1}|^2}\cdot \vec{r}_{1} = -\frac{11}{89}( 4 \vec{e}_{x} + 3\vec{e}_{y} + 8\vec{e}_{z}) = -\frac{44}{89}\vec{e}_{x} - \frac{33}{89}\vec{e}_{y} -\frac{88}{89}\vec{e}_{z}.}\)
e)
\(\displaystyle{ \vec{r}_{1} \times \vec{r}_{2} = ( 4 \vec{e}_{x} + 3\vec{e}_{y} + 8\vec{e}_{z})\times (2\vec{e}_{x} + 10\vec{e}_{y} + 5\vec{e}_{z}) = \left|\begin{matrix} \vec{e}_{x} & \vec{e}_{y} & \vec{e}_{z} \\ 4 & 3 & 8 \\ 2 & 10 & 5 \end{matrix} \right| = -65\vec{e}_{x} - 4\vec{e}_{y} + 34\vec{e}_{z}. }\)
\(\displaystyle{ \vec{r}_{2} = 2\vec{e}_{x} + 10\vec{e}_{y} + 5\vec{e}_{z}. }\)
a)
\(\displaystyle{ \vec{r} = \vec{r}_{2,1} = \vec{r}_{2} - \vec{r}_{1} = -2\vec{e}_{x} + 7\vec{e}_{y} - 3\vec{e}_{z}. }\)
b)
\(\displaystyle{ |\vec{r}_{1}| = \sqrt{\vec{r}_{1}\cdot \vec{r}_{1} } = \sqrt{( 4 \vec{e}_{x} + 3\vec{e}_{y} + 8\vec{e}_{z})\cdot ( 4 \vec{e}_{x} + 3\vec{e}_{y} + 8\vec{e}_{z} )} = \sqrt{4^2 + 3^2 + 8^2} = \sqrt{89}.}\)
\(\displaystyle{ |\vec{r}_{2}| = \sqrt{\vec{r}_{2}\cdot \vec{r}_{2} } = \sqrt{( 2 \vec{e}_{x} + 10\vec{e}_{y} + 5\vec{e}_{z})\cdot ( 2\vec{e}_{x} + 10\vec{e}_{y} + 5\vec{e}_{z} )} = \sqrt{2^2 + 10^2 + 5^2} = \sqrt{129}.}\)
\(\displaystyle{ |\vec{r}_{2,1}| = \sqrt{\vec{r}_{2,1}\cdot \vec{r}_{2,1} } = \sqrt{( -2 \vec{e}_{x} + 7\vec{e}_{y} - 3\vec{e}_{z})\cdot ( -2 \vec{e}_{x} + 7\vec{e}_{y} -3 \vec{e}_{z} )} = \sqrt{(-2)^2 + 7^2 + (-3)^2} = \sqrt{62}.}\)
c)
\(\displaystyle{ \cos(|\angle( \vec{r}_{1}, \vec{r}_{2})|)= \frac{\vec{r}_{1}\cdot \vec{r}_{2}}{|\vec{r}_{1}|\cdot |\vec{r}_{2}|} = \frac{(4 \vec{e}_{x} + 3\vec{e}_{y} + 8\vec{e}_{z})\cdot (2\vec{e}_{x} + 10\vec{e}_{y} + 5\vec{e}_{z})}{\sqrt{89}\cdot \sqrt{129}} = \frac{8 + 30 +40}{\sqrt{89\cdot 129}} = \frac{78}{\sqrt{11392}} \approx 0,73.}\)
\(\displaystyle{ |\angle( \vec{r}_{1}, \vec{r}_{2})|= 43^{o}. }\)
\(\displaystyle{ \cos(|\angle( \vec{r}_{1}, \vec{r}_{2,1})|)= \frac{\vec{r}_{1}\cdot \vec{r}_{2,1}}{|\vec{r}_{1}|\cdot |\vec{r}_{2,1}|} = \frac{(4 \vec{e}_{x} + 3\vec{e}_{y} + 8\vec{e}_{z})\cdot (\vec{r}_{2} - \vec{r}_{1} = -2\vec{e}_{x} + 7\vec{e}_{y} - 3\vec{e}_{z})}{\sqrt{89}\cdot \sqrt{62}} = \frac{-8 + 21 -24}{\sqrt{89\cdot 62}} = \frac{-11}{\sqrt{ 5518}} \approx- 0,15.}\)
\(\displaystyle{ |\angle( \vec{r}_{1}, \vec{r}_{2,1})|= 98,6^{o}. }\)
\(\displaystyle{ \cos(|\angle( \vec{r}_{2}, \vec{r}_{2,1})|)= \frac{\vec{r}_{2}\cdot \vec{r}_{2,1}}{|\vec{r}_{2}|\cdot |\vec{r}_{2,1}|} = \frac{( 2\vec{e}_{x} + 10\vec{e}_{y} + 5\vec{e}_{z})\cdot ( -2\vec{e}_{x} + 7\vec{e}_{y} - 3\vec{e}_{z})}{\sqrt{129}\cdot \sqrt{62}} = \frac{-4 + 70 -15}{\sqrt{129\cdot 62}} = \frac{51}{\sqrt{ 7998}} \approx 0,57 .}\)
\(\displaystyle{ |\angle( \vec{r}_{2}, \vec{r}_{2,1})|= 55,2^{o}. }\)
d)
\(\displaystyle{ rzut (\vec{r}_{2,1})_{\vec{r}_{1}} = \frac{\vec{r}_{2,1}\cdot \vec{r}_{1}}{|\vec{r}_{1}|^2}\cdot \vec{r}_{1} = -\frac{11}{89}( 4 \vec{e}_{x} + 3\vec{e}_{y} + 8\vec{e}_{z}) = -\frac{44}{89}\vec{e}_{x} - \frac{33}{89}\vec{e}_{y} -\frac{88}{89}\vec{e}_{z}.}\)
e)
\(\displaystyle{ \vec{r}_{1} \times \vec{r}_{2} = ( 4 \vec{e}_{x} + 3\vec{e}_{y} + 8\vec{e}_{z})\times (2\vec{e}_{x} + 10\vec{e}_{y} + 5\vec{e}_{z}) = \left|\begin{matrix} \vec{e}_{x} & \vec{e}_{y} & \vec{e}_{z} \\ 4 & 3 & 8 \\ 2 & 10 & 5 \end{matrix} \right| = -65\vec{e}_{x} - 4\vec{e}_{y} + 34\vec{e}_{z}. }\)