1 Four points have coordinates \(\mathrm { A } ( - 2 , - 3,2 ) , \mathrm { B } ( - 3,1,5 ) , \mathrm { C } ( k , 5 , - 2 )\) and \(\mathrm { D } ( 0,9 , k )\).
- Find the vector product \(\overrightarrow { \mathrm { AB } } \times \overrightarrow { \mathrm { CD } }\).
- For the case when AB is parallel to CD ,
(A) state the value of \(k\),
(B) find the shortest distance between the parallel lines AB and CD ,
(C) find, in the form \(a x + b y + c z + d = 0\), the equation of the plane containing AB and CD . - When AB is not parallel to CD , find the shortest distance between the lines AB and CD , in terms of \(k\).
- Find the value of \(k\) for which the line AB intersects the line CD , and find the coordinates of the point of intersection in this case.