Ensino Superior ⇒ Vetores: Combinação Linear

[aprendiz123]

Decomponha o vetor [tex3]\bar{w}[/tex3]=(6,1,-1) como soma de dois vetores [tex3]\bar{w1}[/tex3] e [tex3]\bar{w2}[/tex3], com [tex3]\bar{w1}[/tex3] paralelo ao vetor [tex3]\bar{r}[/tex3]=(-4,4,-2) e [tex3]\bar{w2}[/tex3] ortogonal a [tex3]\bar{r}[/tex3]




[tex3]\,[/tex3]

[Karl Weierstrass]

Decomponha o vetor [tex3]\vec{w}=(6,\,1,\,-1)[/tex3] como soma de dois vetores [tex3]\vec{w}_1[/tex3] e [tex3]\vec{w}_2[/tex3], com [tex3]\vec{w}_1[/tex3] paralelo ao vetor [tex3]\vec{r}\,=\,(-4,\,4,\,-2)[/tex3] e [tex3]\vec{w}_2[/tex3] ortogonal a [tex3]\vec{r}.[/tex3]

Sejam [tex3]\vec{w}_1\,=\,(x_1,\,y_1,\,z_1)[/tex3] e [tex3]\vec{w}_2\,=\,(x_2,\,y_2,\,z_2).[/tex3]

[tex3]\hspace{70pt}\vec{w}_1\,\parallel\,\vec{r}\,\Longrightarrow\,\vec{w}_1\,=\,\alpha\,\cdot\,\vec{r},\,\,\alpha\in\mathbb{R}[/tex3]

[tex3]\hspace{120pt}\,\Longrightarrow\,(x_1,\,y_1,\,z_1)\,=\,(-4\alpha,\,4\alpha,\,-2\alpha)[/tex3]

[tex3]\hspace{120pt}\,\Longrightarrow\,\begin{cases}x_1\,=\,-4\alpha\\ y_1\,=\,4\alpha\\z_1\,=\,-2\alpha\end{cases}[/tex3]

[tex3]\hspace{70pt}\vec{w}\,=\,\vec{w}_1\,+\,\vec{w}_2\,\Longrightarrow\,(6,\,1,\,-1)\,=\,(x_1\,+\,x_2,\,y_1\,+\,y_2,\,z_1\,+\,z_2)[/tex3]


[tex3]\hspace{180pt}\,\Longrightarrow\,\begin{cases}x_1\,+\,x_2\,=\,6\\

y_1\,+\,y_2\,=\,1\\

z_1\,+\,z_2\,=\,-1\end{cases}[/tex3]

[tex3]\hspace{180pt}\,\Longrightarrow\,\begin{cases}x_2\,=\,6\,+\,4\alpha\\

y_2\,=\,1\,-\,4\alpha\\

z_2\,=\,-1\,+\,2\alpha\end{cases}[/tex3]


[tex3]\hspace{70pt}\vec{w}_2\,\perp\,\vec{r}\,\Longrightarrow\,\vec{w}_2\,\cdot\,\vec{r}\,=\,0[/tex3]

[tex3]\hspace{129pt}\,\Longrightarrow\,(x_2,\,y_2,\,z_2)\,\cdot\,(-4,\,4,\,-2)\,=\,0[/tex3]

[tex3]\hspace{129pt}\,\Longrightarrow\,-4x_2\,+\,4y_2\,-\,2z_2\,=\,0[/tex3]

[tex3]\hspace{129pt}\,\Longrightarrow\,-4\,\cdot\,(6\,+\,4\alpha)\,+\,4\,\cdot\,(1\,-\,4\alpha)\,-\,2\,\cdot\,(-1\,+\,2\alpha)\,=\,0[/tex3]

[tex3]\hspace{129pt}\,\Longrightarrow\,\alpha\,=\,-\Large\frac{1}{2}\large.[/tex3]

Portanto,

[tex3]\hspace{70pt}\vec{w}\,=\,\vec{w}_1\,+\,\vec{w}_2\,=\,(2,\,-2,\,1)\,+\,(4,\,3,\,-2).[/tex3]



[tex3]\,[/tex3]

[aprendiz123]

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