Finding the Parametric Equation of a line

Question

Find the parametric equation of the line which contains the point ( 1/2, 0, −3) and
is parallel to the vector 2i − 4j + k. Also, find another point on this line which is
different from ( 1/2, 0, −3).

Answer 1

A vector equation of the line that passes through the point  X₀ = (1/2,0,-3)  and is parallel to the vector  V = 2i - 4j + k  is given by  X = X₀ + tV.  Writing  X = (x,y,z)  gives the parametric equation  

 

(x,y,z) = (1/2,0,-3) + t(2,-4,1) = (1/2+2t,-4t,-3+t)  

 

or the 3 parametric equations  x = 1/2 + 2t,  y = -4t,  z = -3 + t  

 

To find another point on the line pick any non-zero value for t.  For example  

 

t = 1  ==>  (5/2,-4,-2) is a point on the line