Let $A=(1,2,0), B=(2,0,-1), C=(0,-2,3)$ and $D=(-1,2,-3)$ be four points in the space. Let $G_1$ be the centroid of $\triangle A B C$ and $G_2$ be the centroid of tetrahedron $A B C D$. If $P$ divides, $G_1 G_2$ in the ratio $4: 3$ internally, then $P=$

If the d.r.'s of two lines are connected by the relations $a-b+c=0, a^2-b^2+2 c^2=0$ and $\theta$ is the angle between these lines, then $\cos \theta=$

If $l, m$ and $n$ are the d.c.'s of a normal to the plane passing through the points $(0,1,2)$, $(3,0,2)$ and $(4,5,0)$, then $|I|+|m|+|n|=$

Let $L$ be a line passing through the points $2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+8 \hat{\mathbf{k}}$ and $\hat{\mathbf{i}}+6 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}$. Let $P$ be a plane passing through $-5 \hat{\mathbf{i}}+19 \hat{\mathbf{j}}-14 \hat{\mathbf{k}}$ and parallel to the vectors $\hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}$ and $\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}$. If $L$ meets the plane $P$ at a point $A$, then the position vector of $A$, is