Question

Sol. [Key Concept: Solve $\lambda,\mu$ from the two dot-product conditions, then compute ${|c|}^{2}$.]

$$\overrightarrow{c} = \lambda\overrightarrow{a} + \mu\overrightarrow{b}$$

$$\begin{matrix} & & \overrightarrow{c} = (\mu - \lambda)\widehat{i} + (\lambda + 3\mu)\widehat{j} + (3\lambda + \mu)\widehat{k} \\ & & \overrightarrow{c} \cdot (3\widehat{i} - 6\widehat{j} + 2\widehat{k}) = 10 \\ & & 3(\mu - \lambda) - 6(\lambda + 3\mu) + 2(3\lambda + \mu) = 10 \\ & \ - 3\lambda - 13\mu = 10 & \text{(1)} \\ & \overrightarrow{c} \cdot (\widehat{i} + \widehat{j} + \widehat{k}) = - 2 & \text{(1)} \\ & 5\mu + 3\lambda = - 2 & \text{(2)} \end{matrix}$$

Solving (1) & (2)

$${\lambda = 1,\mu = - 1 }{\overrightarrow{c} = - 2\widehat{i} - 2\widehat{j} + 2\widehat{k} }{|\overrightarrow{c}|^{2} = 12 }$$