Question

Find he point of intersection of AB and CD , where $A(6,-7,0),B(16,-19,-4),C(0,3,-6)andD(2,-5,10)$.

Answer 1

Let AB and CD intersect at P.
Let P divides AB in ratio $\lambda :1$ and CD in ratio $\mu :1$.
Then coordinates of P are $(\frac{16\lambda +6}{\lambda +1},\frac{-19\lambda -7}{\lambda +1},\frac{-4\lambda }{\lambda +1})$ or $(\frac{2\mu }{\mu +1},\frac{-5\mu +3}{\mu +1},\frac{10\mu -6}{\mu +1})$
comparing we have $\lambda =-\frac{1}{3}$ or$\mu =1$
Using these values, we get point of intersection as (1, -1, 2).
Here it is also proved that lines AB and CD intersect or points A,B,C and D are coplanar