Question

Dados os vetores U=(-2,1,4) e V(2,1-1), o produto vetorial entre eles e o módulo são respectivamente:

Answer 1

$\mathtt{\overrightarrow{\mathtt{u}}=(-2,\,1,\,4)}\\\\ \mathtt{\overrightarrow{\mathtt{v}}=(2,\,1,\,-1)}$


•    Para o cálculo do produto vetorial, usamos a notação de um determinante simbólico:

$\mathtt{\overrightarrow{\mathtt{u}}\times \overrightarrow{\mathtt{v}}}\\\\ \mathtt{=(-2,\,1,\,4)\times (2,\,1,\,-1)}\\\\\\ =\left| \begin{array}{ccc} \overrightarrow{\mathtt{i}}&\overrightarrow{\mathtt{j}}&\overrightarrow{\mathtt{k}}\\ -2&1&4\\ 2&1&-1 \end{array} \right|\\\\\\ =\left|\begin{array}{cc} 1&4\\ 1&-1 \end{array}\right|\overrightarrow{\mathtt{i}}-\left|\begin{array}{cc} -2&4\\ 2&-1 \end{array}\right|\overrightarrow{\mathtt{j}}+\left|\begin{array}{cc} -2&1\\ 2&1 \end{array}\right|\overrightarrow{\mathtt{k}}$

$=\mathtt{(1\cdot (-1)-1\cdot 4)\overrightarrow{\mathtt{i}}-((-2)\cdot (-1)-2\cdot 4)\overrightarrow{\mathtt{j}}+(-2\cdot 1-2\cdot 1)\overrightarrow{\mathtt{k}}}\\\\ =\mathtt{(-1-4)\overrightarrow{\mathtt{i}}-(2-8)\overrightarrow{\mathtt{j}}+(-2-2)\overrightarrow{\mathtt{k}}}\\\\ =\mathtt{-5\overrightarrow{\mathtt{i}}+6\overrightarrow{\mathtt{j}}-4\overrightarrow{\mathtt{k}}}\\\\ =\mathtt{(-5,\,6,\,-4)}$

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•   O módulo é

$\mathtt{\|\overrightarrow{\mathtt{u}}\times \overrightarrow{\mathtt{v}}\|}\\\\ =\mathtt{\|(-5,\,6,\,-4)\|}\\\\ =\mathtt{\sqrt{(-5)^2+6^2+(-4)^2}}\\\\ =\mathtt{\sqrt{25+36+16}}\\\\ =\mathtt{\sqrt{77}}$


Dúvidas? Comente.


Bons estudos! :-)

Answer 2

Resposta:√77

Explicação passo a passo: