If the volume of the parallelopiped whose conterminus edges are along the vectors $$\mathbf{a}, \mathbf{b}, \mathbf{c}$$ is 12, then the volume of the tetrahedron whose conterminus edges are $$\mathbf{a}+\mathbf{b}, \mathbf{b}+\mathbf{c}$$ and $$c+a$$ is

For any non-zero vectors $$\mathbf{a}$$ and $$\mathbf{b}$$,

If the vectors $$\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}, \hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}$$ and $$2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+m \hat{\mathbf{k}}$$ are coplanar, then $$m=$$

The angles between the lines $$\mathbf{r}=(\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}})+\lambda(\hat{\mathbf{i}}+\hat{\mathbf{j}}+2 \hat{\mathbf{k}}) \text { and } \mathbf{r}=(3 \hat{\mathbf{i}}+\hat{\mathbf{k}})+\lambda^{\prime}(2 \hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}}), \lambda, \lambda^{\prime} \in \mathbf{R}$$ is