Step-by-Step Solution
Key Concept: Multiple dot product constraints form a system that determines both the parameter $\lambda$ and vector relationships through algebraic elimination.
From $\vec{a}\cdot\vec{c}=\frac{1}{4}$ and $\vec{a}\cdot\vec{b}-2\vec{a}\cdot\vec{c}=\lambda\vec{a}\cdot\vec{b}\lambda=\frac{1}{2}$, we solve for relationships. Using $\vec{b}\cdot\vec{b}-2\vec{b}\cdot\vec{c}=\lambda\vec{a}\cdot\vec{b}\vec{c}=8-\frac{k}{2}-\frac{\lambda}{4}$ and the constraint $\vec{b}\cdot\vec{c}-2\vec{c}\cdot\vec{c}=\lambda\vec{a}\cdot\vec{c}$, we obtain $\lambda^2+\lambda-12=0$, giving $\lambda=3,-4$.
Correct Answer: [A-p, q, r, s] [B-p, q] [C-p, r] [D-r]