Determine as projeções A sobre B e de B sobre A: a) A: (2,-1)e B: (-1,1) b) A: (-1,3) e B: (0,4) c0 A: (2,-1,5) e B: (-1,1,1)

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Devemos encontrar as projeções para os três casos dados e para isso realizaremos os cálculos abaixo:

a) \(\begin{align} & pro{{j}_{b}}a=\left( \frac{b\cdot a}{|b{{|}^{2}}} \right)b \\ & pro{{j}_{b}}a=\left( \frac{(-1,1)\cdot (2,-1)}{2} \right)(-1,1) \\ & pro{{j}_{b}}a=\left( \frac{(-2-1)}{2} \right)(-1,1)=\left( \frac{3}{2},\frac{-3}{2} \right) \\ & \\ & pro{{j}_{a}}b=\left( \frac{b\cdot a}{|a{{|}^{2}}} \right)a \\ & pro{{j}_{a}}b=\left( \frac{(-1,1)\cdot (2,-1)}{5} \right)(2,-1) \\ & pro{{j}_{a}}b=\left( \frac{(-2-1)}{5} \right)(2,-1)=\left( \frac{-6}{5},\frac{3}{5} \right) \\ \end{align} \)

\(\boxed{pro{j_a}b = \left( {\frac{{( - 2 - 1)}}{5}} \right)(2, - 1) = \left( {\frac{{ - 6}}{5},\frac{3}{5}} \right)}\)

b) \(\begin{align} & pro{{j}_{b}}a=\left( \frac{(0,4)\cdot (-1,3)}{2} \right)(0,4) \\ & pro{{j}_{b}}a=\left( \frac{(0+12)}{16} \right)(0,4)=\left( 0,3 \right) \\ & \\ & pro{{j}_{a}}b=\left( \frac{(-1,3)\cdot (0,4)}{10} \right)(-1,3) \\ & pro{{j}_{a}}b=\left( \frac{(12)}{10} \right)(-1,3)=\left( \frac{-12}{10},\frac{36}{10} \right) \\ \end{align} \)

\(\boxed{pro{j_a}b = \left( {\frac{{(12)}}{{10}}} \right)( - 1,3) = \left( {\frac{{ - 12}}{{10}},\frac{{36}}{{10}}} \right)}\)

c) 

\(\begin{align} & pro{{j}_{b}}a=\left( \frac{(-1,1,1)\cdot (2,-1,5)}{3} \right)(-1,1,1) \\ & pro{{j}_{b}}a=\left( \frac{(-2-1+5)}{3} \right)(-1,1,1)=\left( \frac{-2}{3},\frac{2}{3},\frac{2}{3} \right) \\ & \\ & pro{{j}_{a}}b=\left( \frac{2}{30} \right)(2,-1,5)=\left( \frac{4}{30},\frac{-2}{30},\frac{10}{30} \right) \\ \end{align} \)

\(\boxed{pro{j_a}b = \left( {\frac{2}{{30}}} \right)(2, - 1,5) = \left( {\frac{4}{{30}},\frac{{ - 2}}{{30}},\frac{{10}}{{30}}} \right)}\)