If in $\triangle A B C, B=45^{\circ}, a=2(\sqrt{3}+1)$ and area of $\triangle A B C$ is $6+2 \sqrt{3}$ sq. units, then the side $b=$

In a $\triangle A B C$, if $\sin ^2 B=\sin A$ and $2 \cos ^2 A=3 \cos ^2 B$, then the triangle is

$P$ 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