Find the parametric equations for the line through C and the midpoint of AB:

Question

Find the parametric equations for the line through C and the midpoint of AB:
A(−4,−3,−5)   B(4,3,−1)   C(3,−1,−1)

Answer 1

The midpoint of AB is  MP = (0,0,-3)

 

A unit vector from MP to C is   u  =(3,-1,2)/sqrt(14)

 

In coordinate form, the parametric form of the line is  C + t u   or  (3,-1,-1) + t (3,-1,2)/sqrt(14)

 

The parametric equations can be read off of this as

 

x = 3 + (3/sqrt(14) ) t

y = -1 -(1/sqrt(14) t

z = -1 + (2/sqrt(14)) t

 

Where t is the parameter which can take on all real values.  

It is easy to check that  t = -sqrt(14)    gives rise to the midpoint MP

Of course t =0  gives the point C