Let $\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+\hat{\mathbf{k}}, \hat{\mathbf{i}}+\hat{\mathbf{j}}-2 \hat{\mathbf{k}}, 2 \hat{\mathbf{i}}-\hat{\mathbf{j}}-\hat{\mathbf{k}}$ and $\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}$ be the position vectors of four points $A, B, C$ and $D$ respectively. If a point $P$ divides $A B$ in the ratio $2: 1$ internally and a point $Q$ divides $C D$ in the ratio $1: 2$ externally, then the ratio in which the point with position vectors $5 \hat{\mathbf{i}}-6 \hat{\mathbf{j}}-5 \hat{\mathbf{k}}$ divides $P Q$ is

If $\mathbf{a}=\hat{\mathbf{i}}+\hat{\mathbf{j}}, \mathbf{b}=2 \hat{\mathbf{j}}-\hat{\mathbf{k}}$ are two vectors such that $\mathbf{r} \times \mathbf{a}=\mathbf{b} \times \mathbf{a} \cdot \mathbf{r} \times \mathbf{b}=\mathbf{a} \times \mathbf{b}$, then the unit vector in the direction of $\mathbf{r}$ is

If $\mathbf{a} \cdot \mathbf{b} \cdot \mathbf{c}$ are three units vectors such that $\mathbf{a} \times(\mathbf{b} \times \mathbf{c})=\frac{\sqrt{3}}{2} \mathbf{b}+\frac{\mathbf{c}}{\mathbf{2}}$ and $\alpha, \beta$ are the angles between $\mathbf{a}, \mathbf{c}$ and $\mathbf{a}, \mathbf{b}$ respectively, then $\alpha+\beta=$

$P$ is the circumcentre of $\triangle A B C$. If the position vectors of $A, B, C$ and $P$ are $\mathbf{a}, \mathbf{b}, \mathbf{c}, \frac{\mathbf{a}+\mathbf{b}+\mathbf{c}}{4}$ respectively, then the position vector of the orthocentre of this triangle is