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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. Normal equation.

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Planes in three dimensions

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


Normal equation
Normal equation

The normal is perpendicular to any line in the plane

(p,q,r)

n

(x,y,z)

R

A

(a,b,c)

Cartesian equation of a plane

o


Theorems
theorems

  • The graph of every linear equation

    is a plane with normal vector (a,b,c)

    2. Two planes with normal vectors a & bare

    (i) parallel if a and b are parallel

    (ii) orthogonal if a and b are orthogonal


Extra cross product
Extra :Cross product

Definition:

determinant

where

is right-handed

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


Cross product
Cross product

1.

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."


Examples
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


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) , 2) p=(-2,5,-1) ,


Angle projection of a line on a plane
Angle ; projection of a line on a plane conditions:

Projection :

L

Normal

n

1.

AB is the projection of line L on the plane

B

projection

A

2. Angle between a line and the plane

The angle between line L and its projection on the plane


Angle between two planes
Angle between two planes conditions:

Firstfind the angle between two normal to the planes

1.

n2

n1

From dot product

to find the angle theta

2. The angle between two planes is


Find the common perpendicular
Find the common perpendicular conditions:

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)


Find the common perpendicular vector
Find the common perpendicular vector conditions:

  • Technique introduced in the textbook.

    the common perpendicular vector of

and

is


Find the normal to the plane
Find the normal to the plane conditions:

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


5. If a line L has parametric equations conditions:

find the plane contain L and point P=(5,0,2)


Line A conditions:

Line B

1

2

Line A

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


Plane A conditions:

Plane B

5

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


2 conditions:

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

mis not on the line.

  • To find the distance from the point mto 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


3 conditions:

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,


4 conditions:

Point A

Plane B

Given a plane:

and a point:

There are only two possibilities: the point is in the plane,

or it isn’t.

simply substitute the coordinates

of m into the equation of the plane.

To check if it is in 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


5 conditions:

Plane A

Plane B

Given a plane:

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.


5 conditions:

Plane A

Plane B

Given a plane:

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


Finding a common perpendicular conditions:

Read section 13.4 p. 182-3 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:


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