If the coordinates of the points `A ,B ,C ,D` are `(1,2,3),(4,5,7),(-4,3,-6)` and `2,9,2),` then find the angle between `A B` and `C Ddot`

Explanation

`:.` Direction ratios of AB
`(4-1),(5-2),(7-3)= (3,3,4)`
Direction ratios of CD
`(2+4),(9-3),(7-3)=(3,3,4)`
Direction of CD
`(2+4),(9-3),(2+6)=(6,6,8)`
Let the directi on ratios of AB and CD are `(a_(1),b_(1),c_(1))` and `(a_(2),b_(2),c_(2))` respectively.
Then, `(a_(1))/(a_(2)) = 3/6 = 1/2 , (b_(1))/(b_(2)) = 3/6 = 1/2 , (c_(1))/(c_(2)) = 4/8 = 1/2`
`(a_(1))/(a_(2)) = (b_(1))/(b_(2)) = (c_(1))/(c_(2))`
`:. (a_(1))/(a_(2)) = (b_(1))/(b_(2)) = (c_(1))/(c_(2))`
Therefore, line AB is parallel to line CD, so the angle between AB and CD is `0^(@)`.