Ensino Superior ⇒ Vetores: Produto Escalar e Produto Vetorial Tópico resolvido

[aprendiz123]

Determine o vetor v que satisfaça as condiçoes:
[tex3]\bar{V}[/tex3].(3 [tex3]\bar{i}[/tex3]+2 [tex3]\bar{j}[/tex3]) = 6
[tex3]\bar{V}[/tex3] x(2 [tex3]\bar{j}[/tex3]+3 [tex3]\bar{k}[/tex3]) = 2 [tex3]\bar{i}[/tex3]


Resposta:
[tex3]\bar{V}[/tex3] = 3 [tex3]\bar{j}[/tex3] + ([tex3]\frac{7}{2}[/tex3]) [tex3]\bar{k}[/tex3]

[Karl Weierstrass]

Determine o vetor [tex3]\vec{v}[/tex3] que satisfaça as condições:

[tex3]\hspace{70pt}\vec{v}\,\cdot\,\(3\vec{i}\,+\,2\vec{j}\)\, =\, 6[/tex3]

[tex3]\hspace{70pt}\vec{v}\,\times\,(2\vec{j}\,+\,3\vec{k}) \,=\, 2\vec{\,i}[/tex3]

Seja [tex3]\vec{v}\,=\,(a,\,b,\,c)[/tex3]

[tex3]\hspace{70pt}\vec{v}\,\cdot\,\(3\vec{i}\,+\,2\vec{j}\)\, =\, 6\,\Longrightarrow\,(a,\,b,\,c)\,\cdot\,(3,\,2,\,0)\, =\, 6\,\Longrightarrow\,3a\,+\,2b\,=\,6[/tex3]

[tex3]\hspace{70pt}\vec{v}\,\times\,(2\vec{j}\,+\,3\vec{k}) \,=\, 2\vec{\,i}\,\Longrightarrow\,\left|\begin{array}{ccc}

\vec{i}& \vec{j}& \vec{k}\\

a& b& c\\

\,&\,&\\

0& 2& 3\\

\end{array}\right|\,=\,2\vec{\,i}\,\Longrightarrow\,(3b\,-\,2c)\vec{i}\,-\,3a\vec{j}\,+\,2a\vec{k}\,=\,2\vec{\,i}[/tex3]

[tex3]\hspace{218pt}\,\Longrightarrow\,3b\,-\,2c\,=\,2[/tex3] e [tex3]a\,=\,0[/tex3]

Mas

[tex3]\hspace{70pt}a\,=\,0\,\Longrightarrow\,3\,\cdot\,0\,+\,2b\,=\,6\,\Longrightarrow\,b\,=\,3[/tex3]

e, portanto,

[tex3]\hspace{70pt}3\,\cdot\,3\,-\,2c\,=\,2\,\Longrightarrow\,c\,=\,\frac{7}{2}[/tex3]



[tex3](*)\,\,[/tex3] [tex3]\vec{a}\,=\,(x_1,\,y_1,\,z_1)[/tex3] e [tex3]\vec{b}\,=\,(x_2,\,y_2,\,z_2)[/tex3]

[tex3]\vec{a}\,=\,\vec{b}\,\Longrightarrow\,\begin{cases}x_1\,=\,x_2\\y_1\,=\,y_2\\z_1\,=\,z_2\end{cases}[/tex3]




[tex3]\,[/tex3]

[aprendiz123]

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