1. **Enunciado do problema:** Dado o referencial ortonormado e os pontos $A(-1,1,2)$, $B(2,-1,6)$ e o vetor $\vec{u} = (4,0,-1)$, determine: 73.1. O ponto $P = B + 2\vec{AB} + \vec{u}$; 73.2. O vetor $\vec{v} = -\frac{1}{2}(3\vec{u}) - \vec{BA}$; 73.3. O vetor $\vec{w}$ sabendo que $2\vec{u} = \frac{1}{2}\vec{AB} + 2\vec{w}$. --- 2. **Fórmulas e regras importantes:** - O vetor $\vec{AB} = B - A$ é calculado subtraindo as coordenadas de $A$ das de $B$. - O vetor $\vec{BA} = A - B = -\vec{AB}$. - Soma e multiplicação de vetores são feitas componente a componente. --- 3. **Cálculo do vetor $\vec{AB}$:** $$\vec{AB} = (2 - (-1), -1 - 1, 6 - 2) = (3, -2, 4)$$ 4. **Cálculo do ponto $P = B + 2\vec{AB} + \vec{u}$:** $$2\vec{AB} = 2 \times (3, -2, 4) = (6, -4, 8)$$ $$P = (2, -1, 6) + (6, -4, 8) + (4, 0, -1)$$ $$P = (2 + 6 + 4, -1 - 4 + 0, 6 + 8 - 1) = (12, -5, 13)$$ --- 5. **Cálculo do vetor $\vec{v} = -\frac{1}{2}(3\vec{u}) - \vec{BA}$:** Primeiro, calcule $3\vec{u}$: $$3\vec{u} = 3 \times (4, 0, -1) = (12, 0, -3)$$ Agora, multiplique por $-\frac{1}{2}$: $$-\frac{1}{2}(3\vec{u}) = -\frac{1}{2} (12, 0, -3) = (-6, 0, \frac{3}{2})$$ Calcule $\vec{BA} = A - B$: $$\vec{BA} = (-1 - 2, 1 - (-1), 2 - 6) = (-3, 2, -4)$$ Agora, calcule $\vec{v}$: $$\vec{v} = (-6, 0, \frac{3}{2}) - (-3, 2, -4) = (-6 + 3, 0 - 2, \frac{3}{2} + 4) = (-3, -2, \frac{11}{2})$$ --- 6. **Cálculo do vetor $\vec{w}$ sabendo que $2\vec{u} = \frac{1}{2}\vec{AB} + 2\vec{w}$:** Rearranjando para $\vec{w}$: $$2\vec{w} = 2\vec{u} - \frac{1}{2}\vec{AB}$$ $$\vec{w} = \frac{1}{2} \left( 2\vec{u} - \frac{1}{2}\vec{AB} \right)$$ Calcule $2\vec{u}$: $$2\vec{u} = 2 \times (4, 0, -1) = (8, 0, -2)$$ Calcule $\frac{1}{2}\vec{AB}$: $$\frac{1}{2} \times (3, -2, 4) = \left( \frac{3}{2}, -1, 2 \right)$$ Subtraia: $$2\vec{u} - \frac{1}{2}\vec{AB} = (8 - \frac{3}{2}, 0 - (-1), -2 - 2) = \left( \frac{16}{2} - \frac{3}{2}, 1, -4 \right) = \left( \frac{13}{2}, 1, -4 \right)$$ Finalmente, divida por 2: $$\vec{w} = \frac{1}{2} \times \left( \frac{13}{2}, 1, -4 \right) = \left( \frac{13}{4}, \frac{1}{2}, -2 \right)$$ --- **Resposta final:** - $P = (12, -5, 13)$ - $\vec{v} = (-3, -2, \frac{11}{2})$ - $\vec{w} = \left( \frac{13}{4}, \frac{1}{2}, -2 \right)$