$P$ 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

If the position vectors of $A, B, C, D$ are $\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+2 \hat{\mathbf{k}}, 2 \hat{\mathbf{i}}-\hat{\mathbf{j}}, \hat{\mathbf{i}}+\hat{\mathbf{j}}+3 \hat{\mathbf{k}}$ and $4 \hat{\mathbf{j}}+5 \hat{\mathbf{k}}$ respectively, then the quadrilateral $A B C D$ is a

The set of all real values of $c$ so that the angle between the vectors $\mathbf{a}=c x \hat{\mathbf{i}}-6 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}$ and $\mathbf{b}=x \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+2 c x \hat{\mathbf{k}}$ is an obtuse angle for all real $x$ is

Let $\mathbf{a}=2 \hat{\mathbf{i}}+\hat{\mathbf{j}}+3 \hat{\mathbf{k}}, \mathbf{b}=3 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+\hat{\mathbf{k}}$ and $\mathbf{c}=\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}$ be three vectors. If $\mathbf{r}$ is a vector such that $\mathbf{r} \times \mathbf{a}=\mathbf{r} \times \mathbf{b}$ and $\mathbf{r} \cdot \mathbf{c}=18$, then the magnitude of the orthogonal projection of $4 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}-\hat{\mathbf{k}}$ on $\mathbf{r}$ is