1. Lines and Planes in Space

2. Lines in Space

4. Vector Equation of a line

7. Example 1
Find a vector equation and parametric equations for the line that passes through the point (5, 1, 3) and is parallel to the vector i + 4 j – 2 k.
Find two other points on the line.

8. Do lines have a unique vector (parametric) equation?

9. Remark
The vector equation and parametric equations of a line are not unique.
If we change the point or the parameter or choose a different parallel vector, then the equations change.

10. Example 2
Find parametric equations and symmetric equations of the line that passes through the points A(2, 4, –3) and B(3, –1, 1).
At what point does this line intersect the xy-plane?

11. Parallel Perpendicular skew lines

12. When do we call two lines are
Parallel?
perpendicular?
Skew?

13. Example-3

14. Example 4
Show that the lines L1 and L2 with parametric equations x = 1 + t y = –2 + 3t z = 4 – t
x = 2s y = 3 + s z = –3 + 4s
are skew lines.
That is, they do not intersect and are not parallel, and therefore do not lie in the same plane.

15. Class work
Homework………. Finish off Class work!