Question

O vetor w= (5,0,12) é a combinação linear dos vetores u= (1,-1,3) e v=(1,4,0). podemos afirmar que os coeficientes a e b dessas combinação são:

Answer 1

Olá

Resposta, a = 4 ;  b = 1

$\vec{u}=(1,-1,3)\\\vec{v}=(1,4,0)\\\vec{w}=(5,0,12)\\\\\\\vec{w}=a\cdot\vec{u}~+~b\cdot\vec{v}\\\\\\(5,0,12)=a(1,-1,3)~+~b(1,4,0)\\\\(5,0,12)=(a,-a,3a)~+~(b,4b,0)\\\\\\\text{Monta um sistema com as respectivas coordenadas dos vetores (x,y,z)}\\\\\\a+b=5~~~~~~~ ~~~~ ~~~~~ ~~~ ~~ ~~~~\longleftarrow\text{Coordenadas 'x' dos vetores}\\-a+4b=0~~~~~~~ ~~~~ ~~~~ ~~~~~~\longleftarrow\text{Coordenadas 'y' dos vetores}\\3a=12~~~~~~~ ~~~~ ~~~~ ~~~~~~~~~ ~~ ~~\longleftarrow\text{Coordenadas 'z' dos vetores}~$

Podemos facilmente encontrar o valor do 'a' na 3ª equação

$\displaystyle 3a=12\\\\a= \frac{12}{3} \\\\\boxed{a=4}\\\\\\\text{Substituindo o valor do 'a' na 1\ª equacao}\\\\a+b=5\\4+b=5\\\\\boxed{b=1}$



Vamos confirmar os valores que encontramos


$(5,0,12)=a(1,-1,3)~+~b(1,4,0)\\\\(5,0,12)=4(1,-1,3)~+~1(1,4,0)\\\\(5,0,12)=(4,-4,12)~+~(1,4,0)\\\\(5,0,12)=(4+1,-4+4,12+0)\\\\\boxed{(5,0,12)=(5,0,12)}~~~~ \checkmark\\\\\\\text{Os valores de 'a'e 'b' que encontramos est\~ao corretos}$