$\mathbf{a}=\hat{\mathbf{i}}+\hat{\mathbf{j}}-2 \hat{\mathbf{k}}, \mathbf{b}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-3 \hat{\mathbf{k}}, \mathbf{c}=2 \hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}$ are three vectors. If $\mathbf{r}$ is a vector such that $\mathbf{r} \cdot \mathbf{a}=0, \mathbf{r} \cdot \mathbf{c}=3$ and $\left[\begin{array}{ll}\mathbf{r} & \mathbf{a} \\ \mathbf{b}\end{array}\right]=0$, then $|\mathbf{r}|=$

In a right angled triangle, if the position vector of the vertex having the right angle is $-3 \hat{\mathbf{i}}+5 \hat{\mathbf{j}}+2 \hat{\mathbf{k}}$ and the position vector of the mid-point of its hypotenuse is $6 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+5 \hat{\mathbf{k}}$, then the position vector of its centroid is

If the position vectors of the vertices $A, B, C$ of a triangle are $3 \hat{\mathbf{i}}+4 \hat{\mathbf{j}}-\hat{\mathbf{k}}, \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+\hat{\mathbf{k}}, 5(\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}})$ respectively, then the magnitude of the altitude drawn from $A$ on to the side $B C$ is

If the vectors $2 \hat{\mathbf{i}}+4 \hat{\mathbf{j}}-3 \hat{\mathbf{k}},-\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}$ and $p \hat{\mathbf{i}}-2 \hat{\mathbf{j}}+\hat{\mathbf{k}}$ are coplanar, then the unit vector in the direction of the vector $9 p \hat{\mathbf{i}}-4 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}$ is