If the vectors $$p \hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}, \hat{\mathbf{i}}+q \hat{\mathbf{j}}+\hat{\mathbf{k}}$$ and $$\hat{\mathbf{i}}+\hat{\mathbf{j}}+r \hat{\mathbf{k}}(p \neq q \neq r \neq 1)$$ are coplanar, then the value of $$p q r-(p+q+r)$$ is

If $$\mathbf{a}=\frac{1}{\sqrt{10}}(3 \hat{\mathbf{i}}+\hat{\mathbf{k}}), \mathbf{b}=\frac{1}{7}(2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}-6 \hat{\mathbf{k}})$$, then the value of $$(2 \mathbf{a}-\mathbf{b}) \cdot[(\mathbf{a} \times \mathbf{b}) \times(\mathbf{a}+2 \mathbf{b})]$$ is

$$\mathbf{a}=2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}, \mathbf{b}=\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+\hat{\mathbf{k}}, \mathbf{c}=\hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}}$$ are three vectors. For a vector $$\mathbf{r}$$ with $$\mathbf{r} \times \mathbf{a}=\mathbf{b}$$ and $$\mathbf{r} \cdot \mathbf{c}=3,|\mathbf{r}|$$ is

If $$\mathbf{a}, \mathbf{b}, \mathbf{c}$$ are non-coplanar unit vectors such that $$\mathbf{a} \times(\mathbf{b} \times \mathbf{c})=\frac{\mathbf{b}+\mathbf{c}}{\sqrt{2}}$$, then the angle between $$\mathbf{a}$$ and $$\mathbf{b}$$ is