Let $A$ be the area of in-circle and $A_1, A_2, A_3$ be the area of ex-circles of a triangle. If $A_1=4, A_2=9, A_3=16$, then $A=$

If $3 \hat{\mathbf{i}}-5 \hat{\mathbf{j}}+2 \hat{\mathbf{k}}, 7 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}-4 \hat{\mathbf{k}}, \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}$ and $-7 \hat{\mathbf{i}}-17 \hat{\mathbf{j}}+16 \hat{\mathbf{k}}$ are position vectors of the points $A, B, C$ and $D$ respectively, then the angle between $\mathbf{A B}$ and $\mathbf{C D}$ is

If $A(2 \hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}}), B(\lambda \hat{\mathbf{i}}+5 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}), C(-4 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+2 \hat{\mathbf{k}})$ and $D(-\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}})$ are four points in space such that $\mathbf{A B}=x \mathbf{A C}+y \mathbf{A D}$ for some real number $x \neq 0, y \neq 0$, then $17(\lambda+9)=$

Let a plane $P$ has the points $\hat{\mathbf{i}}, \hat{\mathbf{j}}$ and $\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}$. Let $L$ be the line through the point $A$ and parallel to the vector $\hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}$. If the plane $P$ and line $L$ intersect at a point $B(0,3,2)$ and the distance from $A$ to $B$ is 3 units, then equations of the normal to the plane $P$ through $A$ are