Question

encontre os numeros reais a e B tais que w=au+Bv onde tem -se u=(2,3,-1) v=(-1,2,-3) e w=(0,7,-5)

Answer 1

$w~=~a.u+b.v\\\\\\(0,7,-5)~=~a\,.\,(2,3,-1)~+~b\,.\,(-1,2,-3)\\\\\\(0,7,-5)~=~(2a,3a,-1a)~+~(-1b,2b,-3b)\\\\\\(0,7,-5)~=~(2a-b~,~3a+2b~,~-a-3b)\\\\\\Igualando~coordenada~a~coordenada,~chegamos ~ao~sistema:\\\\\\\left\{\begin{array}{cccc}2a&-~~~b&=&0\\3a&+~~~2b&=&7\\-a&-~~~3b&=&-5\end{array}\right$

$Isolando~"b"~na~1^a~equacao:\\\\\\\boxed{b~=~2a}\\\\\\Substituindo~na~2^a~equacao:\\\\\\3a+2b~=~7\\\\\\3a+2\,.\,(2a)~=~7\\\\\\3a+4a~=~7\\\\\\7a~=~7\\\\\\a~=~\frac{7}{7}\\\\\\\boxed{a~=~1}\\\\\\Substituindo~o~valor~de~a~para~achar~"b":\\\\\\b~=~2a\\\\\\b~=~2~.~1\\\\\\\boxed{b~=~2}$

Vamos agora verificar se o par (a , b) = (1 , 2) verifica todas equações:

$1^a~eq:\\\\2a-b~=~0\\\\2\,.\,1-2~=~0\\\\0~=~0~\boxed{\checkmark}\\\\\\2^a~eq:\\\\3a+2b~=~7\\\\3\,.\,1+2\,.\,2~=~7\\\\3+4~=~7\\\\7~=~7~\boxed{\checkmark}\\\\\\3^a~eq:\\\\-a+3b~=~-5\\\\-1+3\,.\,2~=~-5\\\\-1+6~=~-5\\\\5~=~-5~ \boxed{\times}$

Como podemos ver, a 3ª equação não foi atendida, ou seja, não há números reais "a" e "b" que satisfazem a equação.