Considere os seguintes vetores de \mathbb{R}^{4}: u=\left(\begin{array}{c}-3 \\\\ 2 \\\\ 1 \\\\ 0\end{array}\right), v=\left(\begin{array}{c}4 \\\\ 7 \\\\ -3 \\\\ 2\end{array}\right) e w=\left(\begin{array}{c}5 \\\\ -2 \\\\ 8 \\\\ 1\end{array}\right). Calcule:
(b) 2 u+7 v

Passo 1

O enunciado fornece os vetores em \mathbb{R}^4
u = \begin{pmatrix}-3 \\\\ 2 \\\\ 1 \\\\ 0\end{pmatrix}, \quad v = \begin{pmatrix}4 \\\\ 7 \\\\ -3 \\\\ 2\end{pmatrix}
e pede para calcular 2u + 7v. Vamos usar a definição de multiplicação escalar e soma de vetores em \mathbb{R}^4.

Passo 2

Calculamos primeiro cada múltiplo escalar:
2u = 2 \cdot \begin{pmatrix}-3 \\\\ 2 \\\\ 1 \\\\ 0\end{pmatrix} = \begin{pmatrix}-6 \\\\ 4 \\\\ 2 \\\\ 0\end{pmatrix}, \qquad 7v = 7 \cdot \begin{pmatrix}4 \\\\ 7 \\\\ -3 \\\\ 2\end{pmatrix} = \begin{pmatrix}28 \\\\ 49 \\\\ -21 \\\\ 14\end{pmatrix}.

Passo 3

Agora somamos componente a componente:
2u + 7v = \begin{pmatrix}-6 \\\\ 4 \\\\ 2 \\\\ 0\end{pmatrix} + \begin{pmatrix}28 \\\\ 49 \\\\ -21 \\\\ 14\end{pmatrix} = \begin{pmatrix}-6+28 \\\\ 4+49 \\\\ 2+(-21) \\\\ 0+14\end{pmatrix} = \begin{pmatrix}22 \\\\ 53 \\\\ -19 \\\\ 14\end{pmatrix}.

Resposta

2u + 7v = \begin{pmatrix}22 \\\\ 53 \\\\ -19 \\\\ 14\end{pmatrix}.

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