1. Planes in three dimensions
Normal equation
Cartesian equation of plane
Angle between a line and a plane or between two planes
Determine whether a line intersects or lies in or parallel to a plane
Distance from a point to a plane

3. Normal equation The normal is perpendicular to any line in the plane n
(p,q,r) n
(x,y,z) R A
(a,b,c)
Cartesian equation of a plane o

4. theorems The graph of every linear equation
is a plane with normal vector (a,b,c)
2. Two planes with normal vectors a & b are
(i) parallel if a and b are parallel
(ii) orthogonal if a and b are orthogonal

5. Extra :Cross product Definition: is right-handed where determinant
where x,y,and z are unit vectors, here u x v is always perpendicular to u and v
Therefore we can find the normal of the plane from two vectors on the plane

6. Cross product 1. The answer is "Elephant grape sine-theta."
The magnitude of the cross product is given by
What’s the dot product???
Joke presented on the television sitcom Head of the Class .
The teacher asks:
"What do you get when you cross an elephant and a grape?"
The answer is "Elephant grape sine-theta."

7. Examples
Find an equation of the plane through the point (5,-2,4) with normal vector a=(1,2,3)
Prove that the planes
and are parallel

8. 1) through P (-2,5,-8) with normal vector a=(-1,-4,1)
3. Find an equation of the plane which satisfies the stated conditions:
1) through P (-2,5,-8) with normal vector a=(-1,-4,1)
2) through P (2,5,-6) and parallel to the plane
3) through the points P(3,2,1) ,Q(-1,1,2) ,R(3,-4,1)
4. find the distance from the point P to the given plane
1) p=(2,1,-1) , ) p=(-2,5,-1) ,

10. Angle ; projection of a line on a plane1.
Projection : L
Normal n
AB is the projection of line L on the plane B A
projection
2. Angle between a line and the plane
The angle between line L and its projection on the plane

11. Angle between two planes1.
First find the angle between two normal to the planes n2 n1
From dot product
to find the angle theta
2. The angle between two planes is

12. Find the common perpendicular
1*.cross product uxv is perpendicular to both u and v.
For example: find the common perpendicular to u=(1,2,3) and v=(7,8,9)

13. Find the common perpendicular vector
Technique introduced in the textbook.
the common perpendicular vector of
Beware of the second term: the order of subtrahend is different from the other
and is

14. Find the normal to the plane
Procedure:
1.Find the two vectors lie on the plane
2. find cross product of the two vectors
Example:
find the Cartesian equation of the plane through A(1,2,1), B(2,-1,-4) and C(1,0,-1)
Two vectors on the plane: AB=(-1,3,5) and AC=(0,2,2)
The vector
is perpendicular to both AB and AC
Therefore -4i+2j-2k is one of the normal to the plane
,equation is -4x+2y-2z=-2

15. 5. If a line L has parametric equations
find the plane contain L and point P=(5,0,2)

16. 1 2 4 3 Line A Line A Line B Point B Angle between lines Skew Find the
perp. distance
of the point
from the line
Intersect
B is on the line A
Parallel (- identical?)
Point A
Plane B 4 3
Line A
Plane B
A is in plane
Line intersects the plane
at one point
Line is in the plane B
Find the perp. distance
of the point from the plane
A is parallel to B

17. 5 1 Plane A Plane B Angle between planes There is a line of
Intersection m
Parallel (- identical?) 1
Given the two vector equations of the lines, you can examine the
three simultaneous equations:
If two equations yield a specific
pair of values (t,s) and the third equation
is consistent, then there is an intersection.
If there are no solutions, the lines are skew or parallel. If q is a multiple of p, the lines are parallel, otherwise they are skew.
If there are an infinite number of solutions,
the lines are identical. p θ
The angle between two lines is easily found using the dot product. q

18. 2 To examine whether a point is on a line, simply find a value of the
Parameter t which satisfies the equation for each coordinate:
If there is no value of t which makes
these equations all true, then the point
m is not on the line.
To find the distance from the point m to the line:
calculate the unit normal vector
Find the displacement vector from A to m
Take the dot product of these two vectors
Use Pythagoras to find the third side of the triangle X d A M

19. 3 Line A Plane B Given a plane: and a line:
… the line is parallel to the plane if:
Moreover it is in the plane, also, if:
This must be true for all t, so t must cancel out and the
LHS=RHS= a constant.
If there are no solutions for t:
the line is parallel to the plane, but not in it.
If there is one solution for t:
the line intersects the plane.
In the last case,

20. 4 n Point A Given a plane: Plane B and a point:
There are only two possibilities: the point is in the plane,
or it isn’t.
To check if it is in the plane:
simply substitute the coordinates
of m into the equation of the plane. n
To find the distance of the point from the plane: A
(a,b,c)
The simplest way is to find a unit
normal vector and then calculate: d m

21. 5 Plane A Given a plane: Plane B and a plane:
The planes may be parallel or identical or …?
Intersect in a line n2 n1
This diagram shows how to find the angle between the planes.

22. 5 Plane A Given a plane: Plane B and a plane:
To find the equation of the line:
Choose any two values of x.
Use the two plane equations to find the corresponding y and z
which solves both plane equations
Now you have two points on the line, so you can find its equation

23. Finding a common perpendicular
Read section 13.4 p carefully. The method described is actually the method
to find the cross product.
Find a vector perpendicular to each pair of vectors:
Note that this method is a good way
to find the equation of a plane through
three points A, B, C: