Question

Qual o vetor projeção ortogonal de u⃗ =(9,−1,2) sobre v⃗ =(1,3,4)?

Answer 1

Resposta:

Proj $v^u$= $\frac{( u * v)}{|v|*|v|} * v$  = $(\frac{7}{13} , \frac{21}{13} ,\frac{28}{13} )$

Explicação passo-a-passo:

(u*v) = (9,−1,2) * (1,3,4) = 9 * 1 - 1 * 3 + 2 * 4 = 6 + 8 = 14

|v| = $\sqrt{1^{2} + 3^{2} + 4^{2} } = \sqrt{1+9+16} = \sqrt{26}$

|v| * |v| = $\sqrt{26}$ * $\sqrt{26}$ = 26

$\frac{14}{26} / 2 = \frac{7}{13}$

$\frac{7}{13} * (1,3,4) = (\frac{7}{13}, \frac{21}{13}, \frac{28}{13})$

Confira as respostas!

Answer 2

✅ Após ter realizado os cálculos concluímos que a projeção ortogonal procurada é:

     $\Large\displaystyle\text{ \begin{gathered}proj_{\vec{v}}\:\vec{u} = \Bigg(\frac{7}{13}, \frac{21}{13}, \frac{28}{13} \Bigg) \end{gathered} }$

Sejam os vetores:

           $\Large\begin{cases}\vec{u} = (9, -1, 2)\\\vec{v} = (1, 3, 4) \end{cases}$

A projeção ortogonal de u por v pode ser calculada utilizando a seguinte fórmula:

    $\Large\displaystyle\text{ \begin{gathered}proj_{\vec{v}}\:\vec{u} = \Bigg(\frac{\vec{u}\cdot\vec{v}}{\|\vec{v}\|^{2}} \Bigg)\cdot\vec{v} \end{gathered} }$

                      $\Large\displaystyle\text{ \begin{gathered}= \Bigg(\frac{(9, -1, 2)\cdot(1, 3, 4)}{(\sqrt{1^{2} + 3^{2} + 4^{2}})^{2}} \Bigg)\cdot(1, 3, 4) \end{gathered} }$

                       $\Large\displaystyle\text{ \begin{gathered}= \Bigg(\frac{9\cdot1 + (-1)\cdot3 + 2\cdot4}{1^{2} + 3^{2} + 4^{2}} \Bigg)\cdot(1, 3, 4) \end{gathered} }$

                       $\Large\displaystyle\begin{gathered}= \Bigg(\frac{9 - 3 + 8}{1 + 9 + 16} \Bigg)\cdot(1, 3, 4) \end{gathered}$

                        $\Large\displaystyle\begin{gathered}= \Bigg(\frac{14}{26} \Bigg)\cdot(1, 3, 4) \end{gathered}$

                         $\Large\displaystyle\begin{gathered}= \frac{7}{13}\cdot(1, 3, 4) \end{gathered}$

                         $\Large\displaystyle\begin{gathered}= \Bigg(\frac{7}{13}, \frac{21}{13}, \frac{28}{13} \Bigg) \end{gathered}$

✅ Portanto, a projeção ortogonal é:

        $\Large\displaystyle\text{ \begin{gathered}proj_{\vec{v}}\:\vec{u} = \Bigg(\frac{7}{13}, \frac{21}{13}, \frac{28}{13} \Bigg) \end{gathered} }$

Saiba mais:

1. https://brainly.com.br/tarefa/8071524
2. https://brainly.com.br/tarefa/28955137