If P divides the line segment joining the points $$A(1,2,-1)$$ and $$B(-1,0,1)$$ externally in the ratio 1 : 2 and $$Q=(1,3,-1)$$, then $$PQ=$$

If the direction cosines of a line are $$\left(\frac{a}{\sqrt{83}}, \frac{5}{\sqrt{83}}, \frac{c}{\sqrt{83}}\right)$$ and $$c-a=4$$, then $$ca=$$

Let the plane $$\pi$$ pass through the point (1, 0, 1) and perpendicular to the planes $$2x + 3y - z = 2$$ and $$x - y + 2z = 1$$. Let the equation of the plane passing through the point (11, 7, 5) and parallel to the plane $$\pi$$ be $$ax + by - z - d = 0$$. Then, $${a \over b} + {b \over d} = $$

$$D, E, F$$ are respectively the points on the sides $$B C, C A$$ and $$A B$$ of a $$\triangle A B C$$ dividing them in the ratio $$2: 3,1: 2,3: 1$$ internally. The lines $$\mathbf{B E}$$ and $$\mathbf{C F}$$ intersect on the line $$\mathbf{A D}$$ at $$P$$. If $$\mathbf{A P}=x_1 \cdot \mathbf{A} \mathbf{B}+y_1 \cdot \mathbf{A C}$$, then $$x_1+y_1=$$