1. Chapter 12 – Vectors and the Geometry of Space
12.5 Equations of Lines and Planes
Objectives:
Find vector, parametric, and general forms of equations of lines and planes.
Find distances and angles between lines and planes
12.5 Equations of Lines and Planes

2. Lines in 2D
A line in the xy-plane is determined when a point on the line and the direction of the line (its slope or angle of inclination) are given.
12.5 Equations of Lines and Planes

3. Lines in 3D A line L in 3D space is determined when we know:
A point P0(x0, y0, z0) on L
The direction of L, given by a vector
12.5 Equations of Lines and Planes

4. Definition – Vector Equation
So, we let v be a vector parallel to L.
Let P(x, y, z) be an arbitrary point on L.
Let r0 and r be the position vectors of P0 and P.
If a is the vector with representation from P to Po.
Then the Triangle Law for vector addition gives r = r0 + a.
However, since a and v are parallel vectors, there is a scalar t such that a = tv.
So we have
This is a vector equation of L.
r = r0 + t v 1
12.5 Equations of Lines and Planes

5. Definition continued
We can also write this definition in component form: r = <x, y, z> and r0 = <x0, y0, z0>
So, vector Equation 1 becomes:
<x, y, z> = <x0 + ta, y0 + tb, z0 + tc> 2
Positive values of t correspond to points on L that lie on one side of P0.
Negative values
correspond to points that lie on the other side.
12.5 Equations of Lines and Planes

6. x = x0 + at y = y0 + bt z = z0 + ct, where t ℝ
Definition continued
Two vectors are equal if and only if corresponding components are equal.
Hence, we have the following three scalar equations.
x = x0 + at y = y0 + bt z = z0 + ct, where t ℝ
These equations are called parametric equations of the line L through the point P0(x0, y0, z0) and parallel to the vector v = <a, b, c>.
Each value of the parameter t gives a point (x, y, z) on L.
12.5 Equations of Lines and Planes

7. Example 1 – pg 824 #2
Find a vector equation and parametric equations for the line.
The line through the point (6,-5,2) and parallel to the vector <1,3,-2/3>.
12.5 Equations of Lines and Planes

8. Definition - Symmetric Equations
If we solve the equations for t
x = x0 + at, y = y0 + bt, z = z0 + ct,
we get the following symmetric equations. 3
12.5 Equations of Lines and Planes

9. Example 2
Find parametric equations and symmetric equations for the line.
The line through the points (6,1,-3) and (2,4,5).
12.5 Equations of Lines and Planes

10. Equations of Line Segments
The line segment from r0 to r1 is given by the vector equation
where 0 ≤ t ≤ 1
r(t) = (1 – t)r0 + t r1 4
12.5 Equations of Lines and Planes

11. Definition – Skew Lines
Lines that are skew are not parallel and do NOT intersect. They do NOT lie in the same plane.
12.5 Equations of Lines and Planes

12. Example 3
Determine whether the lines L1 and L2 are parallel, skew, or intersecting. If they intersect, find the point of intersection.
12.5 Equations of Lines and Planes

13. Planes A plane in space is determined by:
A point P0(x0, y0, z0) in the plane
A vector n that is orthogonal to the plane
12.5 Equations of Lines and Planes

14. Definition – Normal Vector
This orthogonal vector n is called a normal vector.
The normal vector n is orthogonal to every vector in the given plane.
In particular, n is orthogonal to r – r0.
12.5 Equations of Lines and Planes

15. Equation of Planes
If n is orthogonal to r – r0 we have the following equations:
Either Equation 5 or Equation 6 is called a vector equation of the plane.
12.5 Equations of Lines and Planes

16. Equations of Planes
This equation is the scalar equation of the plane through P0(x0, y0, z0) with normal vector n = <a, b, c>.
12.5 Equations of Lines and Planes

17. Linear Equation where d = –(ax0 + by0 + cz0)
This is called a linear equation in x, y, and z.
12.5 Equations of Lines and Planes

18. Example 4 – pg. 825 # 34 Find an equation of the plane.
The plane that passes through the point (1,2,3) and contains the line x = 3t, y = 1 + t, z = 2 – t.
12.5 Equations of Lines and Planes

19. Example 5 – pg. 825 # 45
Find the point at which the line intersects the given plane.
x = 3 – t
y = 2 + t
z = 5t
x – y + 2z = 9
12.5 Equations of Lines and Planes

20. Example 6 – pg. 825 #48
Where does the line through (1,0,1) and (4, -2, 2) intersect the plane x + y + z = 6?
12.5 Equations of Lines and Planes

21. More Examples
The video examples below are from section 12.5 in your textbook. Please watch them on your own time for extra instruction. Each video is about 2 minutes in length.
Example 3
Example 4
Example 7
12.5 Equations of Lines and Planes

22. Demonstrations Feel free to explore these demonstrations below.
Constructing Vector Geometry Solutions
12.5 Equations of Lines and Planes