Question

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Determinar o vetor projeção de
U=(2,3,4) sobre V=(1,-1,0)

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Answer 1

$\boxed{\boxed{proj_{\vec A}\vec B= \left(\frac{B.A}{|A|^2 }\right) *A}}\\\\\\\ \bmatrix{ proj_{\vec A}\vec B= \text{projecao do vetor B sobre A}\\\\(B.A) = \text{produto escalar entre o vetor B e A}\\\\ |A|= modulo do vetor A\end$

temos:
queremos U sobre V

$U=(2,3,4)\\\\V=(1,-1,0)\\\\|V|^2=1^2+(-1)^2+0)^2 = 1+1=2\\\\\\\\\\ proj_{\vec V}\vec U= \left (\frac{(2,3,4).(1,-1,0)}{2} \right)*(1,-1,0) \\\\ proj_{\vec V}\vec U= \left (\frac{(2*1)+(3*-1)+(4*0)}{2} \right)*(1,-1,0) \\\\ proj_{\vec V}\vec U= \left (\frac{2-3+0}{2} \right)*(1,-1,0) \\\\ proj_{\vec V}\vec U= \frac{-1}{2}*(1,-1,0)=\left( \frac{-1}{2}, \frac{1}{2},0\right)$