$\mathbf{1}, \mathbf{m}, \mathbf{n}$ are three unit vectors in a right handed system and $L$ is a line through the points $A, B, C$ whose position vectors are $p \mathbf{1}+7 \mathbf{m}-6 \mathbf{n}, 2 \mathbf{1}+5 \mathbf{m}-4 \mathbf{n}$ and $1+4 \mathbf{m}-3 \mathbf{n}$ respectively. If the equation of the plane containing $L$ and the points ( $-p, p, p+1$ ) is $a x+b y+c z=1$, then $p(a+b+c)=$

$E(1,0,0), F(0,2,0), G(0,0,3)$ are respectively the mid-points of the sides $A B, B C, C A$ of $\triangle A B C$. If $a_1, b_1, c_1$ and $a_2, b_2, c_2$ are respectively the direction ratios of $A F$ and $B G$, then $\frac{a_1^2+b_1^2+c_1^2}{a_2^2+b_2^2+c_2^2}=$

If the direction ratios $a, b, c$ of a line $L$ satisfy the relations $a b+b c+c a=0$ and $6 a b+9 b c+8 c a=0$, then the direction cosines of the line $L$ are

The equation of the plane passing through the line of intersection of planes $\pi_1=2 x+6 y+4 z-7=0$, $\pi_2=x-y-2 z-2=03$ and perpendicular to the plane $x+y+2 z-5=0$ is