1. 3D Lines and Planes

2. Direction Numbers
The vector is the direction vector for the line L and a, b, and c are
called the direction numbers.
Line in Three Dimensions

3. Line in Three Dimensions
Parametric Equations:
Line in Three Dimensions
Vector Equation:
Line in Three Dimensions
Cartesian or Symmetric Equations:

4. Find the Cartesian equation of the line which passes through the point (1, -2, 4) and which is parallel to the vector below.
Answer:
Find the Parametric equations of the line which passes through the points (-2, 1, 0) and(1, 3, 5).
Answer:

5. Find the vector equation of the line which passes through the point with position vector i + j + k and which is parallel to the vector below.
Answer:
Find the Parametric equations of the line which passes through the points (2, 3, -1) and(-1, 3, 2).
Answer:

6. Two lines have vector equations given below
Two lines have vector equations given below. Find their point of intersection.
Answer: (-4, 3, 8)

7. Two lines have vector equations given below
Two lines have vector equations given below. Find the angle between these two lines.
Answer: 2.51

8. The vector equations of two lines are given by:
The two lines intersect at the point P. Find the position vector of P.
Answer:
M02/HL1/8

9. Consider the four points A(1, 4, -1), B(2, 5, -2), C(5, 6, 3) and D(8, 8, 4). Find the point of intersection of the lines (AB) and (CD).
Answer: (-1, 2, 1)
N04/HL1/11

10. Planes The equation of a plane can be found using two vectors.
The following plane consists of all
points Q(x, y, z) for which the
vector is orthogonal to a
vector

11. This is called the standard form of an equation of a plane.

12. or
This is another version of the standard form of an equation of a plane.

13. Standard form of an equation of a plane
General or Cartesian form of an equation of a plane

14. Find the Cartesian equation of the plane containing the points (2, 1, 1), (0, 4, 1) and (-2, 1, 4).
Answer:
If A = (1, 0, 2) and B = (3, -4, 6), find a Cartesian equation of the plane perpendicular to AB and through the midpoint of AB.
Answer:

15. Find the Cartesian equation of the plane which passes through the points (2, 2, 1) and (-1, 1, -1) and which is perpendicular to the plane
Answer:
Find the Cartesian equation of the plane which contains the lines given below:
Answer:

16. This is the vector or parametric form of the equation of the plane.

17. Write
in Cartesian form.

18. Angle between Two Planes
The angle between two planes can be found by finding the angle between the two normal vectors defined by the plane.

19. Find the angle between the two planes whose equations are below.
Answer:
Find the angle between the two planes whose equations are below.
Answer:

20. Finding the Line of Intersection of Two Planes
Find the line of intersection between the two planes whose equations are given below.
Answer:
Notice that the Direction numbers for the line of intersection are orthogonal to the normal vectors of the planes.
Notice also that both planes pass through the point (0, 0, 0). Therefore the point (0, 0, 0) lies on the line of intersection.

21. Find an equation for the line of intersection of the two planes:
Answer: N02/HL1/10
Find an equation for the line of intersection of the two planes:
Answer: