Three lines are given by

$$r = \lambda \widehat i,\,\lambda \in R$$,

$$r = \mu (\widehat i + \widehat j),\,\mu \in R$$ and

$$r = v(\widehat i + \widehat j + \widehat k),\,v\, \in R$$

Let the lines cut the plane x + y + z = 1 at the points A, B and C respectively. If the area of the triangle ABC is $$\Delta $$ then the value of (6$$\Delta $$)² equals ..............

Let P be a point in the first octant, whose image Q in the plane x + y = 3 (that is, the line segment PQ is perpendicular to the plane x + y = 3 and the mid-point of PQ lies in the plane x + y = 3) lies on the Z-axis. Let the distance of P from the X-axis be 5. If R is the image of P in the XY-plane, then the length of PR is ...............

Consider the cube in the first octant with sides OP, OQ and OR of length 1, along the X-axis, Y-axis and Z-axis, respectively, where O(0, 0, 0) is the origin. Let $$S\left( {{1 \over 2},{1 \over 2},{1 \over 2}} \right)$$ be the centre of the cube and T be the vertex of the cube opposite to the origin O such that S lies on the diagonal OT. If p = SP, q = SQ, r = SR and t = ST, then the value of |(p $$ \times $$ q) $$ \times $$ (r $$ \times $$ t)| is ............

If the distance between the plane $$Ax-2y+z=d$$ and the plane containing the lines $${{x - 1} \over 2} = {{y - 2} \over 3} = {{z - 3} \over 4}$$ and $${{x - 2} \over 3} = {{y - 3} \over 4} = {{z - 4} \over 5}\,$$ is $$\sqrt 6 \,\,,$$ then find $$\left| d \right|.$$