1. Notation of lines in space
Vector equation
Parametric equations
This can be made up from the vector equation:
Make up the equations for y and z.
coordinate on the line
the vector part

2. Lines in space - questions
1. A line, L1, parallel to the vector 4i+2j-k passes through the coordinate (1,2,3).
a) Write down the vector equation of the line L1.
b) Write down the parametric equation of the line L1.
2. A straight line, L2, passes through the points A(2,5,-1) and B(5,-1,0).
a) Find the vector AB.
b) Hence write down the parametric equation of L2.

3. Shortest distance in space
As before with closest distance from a point to a line is by making a path from the point to the line that will meet at a right angle.
Vectors AB and PQ will meet at right angles.
Example
Find the shortest distance from the point P(-5,10,-6) to the line with parametric equations
x=1-3t,y=4+2t,z=1-4t. B Q A
P(-5,10,-6)
If t=2 then Q(-5,8,-7).
Let Q be where the lines meet.
Q will have the coordinates:
(1-3t, 4+2t, 1-4t)