If $\hat{\mathbf{i}}+\hat{\mathbf{j}}, \hat{\mathbf{j}}+\hat{\mathbf{k}}, \hat{\mathbf{k}}+\hat{\mathbf{i}}, \hat{\mathbf{i}}-\hat{\mathbf{j}}, \hat{\mathbf{j}}-\hat{\mathbf{k}}$ are the position vectors of the points $A, B, C, D, E$ respectively, then the point of intersection of the line $A B$ and the plane passing through $C, D, E$ is.

A plane $(\pi)$ passing through the point $(1,2,-3)$ is perpendicular to the planes $x+y-z+4=0$ and $2 x-y+z+1=0$. If the equation of the plane $(\pi)$ is $a x+b y+c z+1=0$, then $a^2+b^2+c^2=$

If the ratio of the perpendicular distances of a variable point $P(x, y, z)$ from the $X$-axis and from the $Y Z$ - plane is $2: 3$, then the equation of the locus of $P$ is

The direction cosines of two lines are connected by the relations $l-m+n=0$ and $2 l m-3 m n+n l=0$. If $\theta$ is the angle between these two lines, then $\cos \theta=$