Find a vector parameterization for the line through P 1 1 1

Find a vector parameterization for the line through P = (1, 1, 1) that is parallel to the line through (2, 0, -1) and (4, 1, 3).

Solution

Line is the parallel to the line through (2,0,-1) and (4,1,3)

So line is parallel to the vector: v=(4,1,3)-(2,0,-1)=(2,1,4)

So the vector parametrisation for the required line is

x=(1,1,1)+t(2,1,4)

Note that setting t=0 gives us the point P so point P lies on this line

ANd take any two points some t=t1,t2

x1=(1,1,1)+t1(2,1,4)

x2=(1,1,1)+t2(2,1,4)

IT is parallel to the vector

x2-x1=(t2-t1)v

Henve this is the correct parametrisation