Let $\mathbf{a}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-2 \hat{\mathbf{k}}$ and $\mathbf{b}=2 \hat{\mathbf{i}}-\hat{\mathbf{j}}-2 \hat{\mathbf{k}}$. If the orthogonal projection vector of $\mathbf{a}$ on $\mathbf{b}$ be $\mathbf{x}$ and the orthogonal projection vector of $\mathbf{b}$ on $\mathbf{a}$ be $\mathbf{y}$, then $|\mathbf{x}-\mathbf{y}|=$

Let $\mathbf{p}, \mathbf{q}, \mathbf{r}$ be three non-coplanar vectors and $\left[\begin{array}{lll}\mathbf{p} & \mathbf{q} & \mathbf{r}] \mathbf{a}=\mathbf{q} \times \mathbf{r},\left[\begin{array}{lll}\mathbf{p} & \mathbf{q} & \mathbf{r}] \mathbf{b}=\mathbf{r} \times \mathbf{p},[ \end{array}[\mathbf{p}\right. \\ \mathbf{q} & \mathbf{r}] \mathbf{c}=\mathbf{p} \times \mathbf{q} \text {. If }\end{array}\right. \mathbf{a}, \mathbf{b}, \mathbf{c}$ denote the coterminous edges of a parallelopiped, then its height with the base having a and $\mathbf{c}$ is

If $\mathbf{b}, \mathbf{c}$ are non collinear vectors, $|\mathbf{c}| \neq 0$, $\mathbf{a} \times(\mathbf{b} \times \mathbf{c})+(\mathbf{a} \cdot \mathbf{b}) \mathbf{b}=(4-2 \beta-\sin \alpha) \mathbf{b}+\left(\beta^2-1\right) \mathbf{c}$ and $(\mathbf{c} \cdot \mathbf{c}) \mathbf{a}=\mathbf{c}$, then the scalars $\alpha$ and $\beta$ are

If $12 \hat{\mathbf{i}}-12 \hat{\mathbf{j}}-18 \hat{\mathbf{k}},-3 \hat{\mathbf{i}}-6 \hat{\mathbf{j}}-9 \hat{\mathbf{k}}$ and $3 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}-24 \hat{\mathbf{k}}$ be the position vectors of the vertices $A, B$ and $C$ respectively of $\triangle A B C$, then the position vector of the incentre of $\triangle A B C$ is