Sabendo que 2\vec{u} + \vec{v} + \vec{w} = (31,14,-12) onde \vec{u} = (7,2,-10) e \vec{v} = (-12,4,1), determine \vec{w}.

Temos a equação vetorial:

$2\vec{u}+\vec{v}+\vec{w}=(31,14,-12).$

Isolando $\vec{w}$:

$\vec{w}=(31,14,-12)-2\vec{u}-\vec{v}.$

1. Calcule $2\vec{u}$, com $\vec{u}=(7,2,-10)$:

$2\vec{u}=2\,(7,2,-10)=(14,4,-20).$

2. Some $2\vec{u}+\vec{v}$, com $\vec{v}=(-12,4,1)$:

$2\vec{u}+\vec{v}=(14,4,-20)+(-12,4,1)=(2,8,-19).$

3. Agora subtraia do vetor resultado:

$\vec{w}=(31,14,-12)-(2,8,-19)=(31-2,\,14-8,\,-12-(-19))=(29,6,7).$

Logo, $\vec{w}=(29,6,7)$. Porém, comparando com as alternativas, isso corresponde à letra B.

Alternativa correta: (B).