Determine a amplitude do ângulo dos dois vetores
Geometria Analítica: Infinito 11 A - Parte 1 Pág. 179 Ex. 15
Enunciado
Determine (se necessário apresente o resultado aproximado às décimas) a amplitude do ângulo de $\overrightarrow{u}$ com $\overrightarrow{v}$, sabendo que:
- $\overrightarrow{u}.\overrightarrow{v}=50\sqrt{2}$ e $\left\| \overrightarrow{u} \right\|=\left\| \overrightarrow{v} \right\|=10$
- $\overrightarrow{u}.\overrightarrow{v}=-10\sqrt{3}$ e $\left\| \overrightarrow{u} \right\|=4$ e $\left\| \overrightarrow{v} \right\|=5$
- $\overrightarrow{u}(1,2,3)$ e $\overrightarrow{v}(-1,1,-1)$
- $\overrightarrow{u}(\frac{2}{3},\frac{2}{3},1)$ e $\overrightarrow{v}(-\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2},0)$
Resolução
- Ora,
\[\begin{array}{*{35}{l}}
\cos (\widehat{\vec{u}\,\vec{v}}) & = & \frac{\vec{u}.\vec{v}}{\left\| {\vec{u}} \right\|.\left\| {\vec{v}} \right\|} \\
{} & = & \frac{50\sqrt{2}}{10\times 10} \\
{} & = & \frac{\sqrt{2}}{2} \\
\end{array}\]
Logo, $\widehat{\vec{u}\,\vec{v}}=45{}^\text{o}$.
- Ora,
\[\begin{array}{*{35}{l}}
\cos (\widehat{\vec{u}\,\vec{v}}) & = & \frac{\vec{u}.\vec{v}}{\left\| {\vec{u}} \right\|.\left\| {\vec{v}} \right\|} \\
{} & = & \frac{-10\sqrt{3}}{4\times 5} \\
{} & = & -\frac{\sqrt{3}}{2} \\
\end{array}\]
Logo, $\widehat{\vec{u}\,\vec{v}}=150{}^\text{o}$.
- Ora,
\[\begin{array}{*{35}{l}}
\cos (\widehat{\vec{u}\,\vec{v}}) & = & \frac{\vec{u}.\vec{v}}{\left\| {\vec{u}} \right\|.\left\| {\vec{v}} \right\|} \\
{} & = & \frac{(1,2,3).(-1,1,-1)}{\sqrt{{{1}^{2}}+{{2}^{2}}+{{3}^{2}}}\times \sqrt{{{(-1)}^{2}}+{{1}^{2}}+{{(-1)}^{2}}}} \\
{} & = & \frac{1\times (-1)+2\times 1+3\times (-1)}{\sqrt{14}\times \sqrt{3}} \\
{} & = & -\frac{2}{\sqrt{14}\times \sqrt{3}} \\
{} & = & -\frac{2\sqrt{42}}{42} \\
{} & = & -\frac{\sqrt{42}}{21} \\
\end{array}\]
Logo, $\widehat{\vec{u}\,\vec{v}}=108,0{}^\text{o}$.
- Ora,
\[\begin{array}{*{35}{l}}
\cos (\widehat{\vec{u}\,\vec{v}}) & = & \frac{\vec{u}.\vec{v}}{\left\| {\vec{u}} \right\|.\left\| {\vec{v}} \right\|} \\
{} & = & \frac{(\frac{2}{3},\frac{2}{3},1).(-\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2},0)}{\sqrt{{{\left( \frac{2}{3} \right)}^{2}}+{{\left( \frac{2}{3} \right)}^{2}}+{{1}^{2}}}\times \sqrt{{{\left( -\frac{\sqrt{2}}{2} \right)}^{2}}+{{\left( \frac{\sqrt{2}}{2} \right)}^{2}}+{{0}^{2}}}} \\
{} & = & \frac{-2\sqrt{2}+2\sqrt{2}+0}{\sqrt{\frac{17}{9}}\times \sqrt{1}} \\
{} & = & 0 \\
\end{array}\]
Logo, $\widehat{\vec{u}\,\vec{v}}=90{}^\text{o}$.
