Question #41933

Let $L_1$ and $L_2$ be two lines in 3D space whose direction ratios are respectively given by $\langle a_1, b_1, c_1 \rangle$ and $\langle a_2, b_2, c_2 \rangle$, where: $a_1 = [ 3 S ], \quad b_1 = [ -2 S ], \quad c_1 = [ 4 S ]$ $a_2 = [ 2 T ], \quad b_2 = [ 6 T ], \quad c_2 = [ 3 T ]$ Here, $[x]$ denotes the greatest integer less than or equal to $x$, and the sums $S$ and $T$ are defined as: $S = \sum_{k=1}^{10} \frac{k}{(k+1)!}$ $T = \sum_{k=1}^{24} \frac{1}{k\sqrt{k+1} + (k+1)\sqrt{k}}$ Find the angle between the lines $L_1$ and $L_2$.