Planes in three dimensions

1. Normal equation2. Cartesian equation of plane3. Angle between a line and a plane or between two planes4. Determine whether a line intersects or lies in or parallel to a plane5. Distance from a point to a plane

Normal equation

The normal is perpendicular to any line in the plane

n

o

A R

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0AR n��������������

( ) 0 n r a

n r n a

(p,q,r)

(a,b,c)

(x,y,z)

px qy rz n aCartesian equation of a plane

px qy rz k

theorems

1. 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

ax by cy d

Extra :Cross product

Definition:

where

is right-handed

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

ˆ ˆ ˆ

x y z

x y z

u u u

v v v

x y z

u v

Cross product

1. The magnitude of the cross product is given by

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."

cos u v u v

What’s the dot

product???sin u v u v

Examples

1. Find an equation of the plane through the point (5,-2,4) with normal vector a=(1,2,3)

2. Prove that the planes

and are parallel

2 3 5 0x y z

6 9 3 2 0x x z

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) ,

3 2 10x y z

3 7 5x y z 4 3 2x z

Angle ; projection of a line on a plane

1.Normal

n

projection

L

A

B

AB is the projection of line L on the plane

The angle between line L and its projection on the plane

2. Angle between a line and the plane

Projection :

Angle between two planes

1.

n1

n2

180

2. The angle between two planes is 180

From dot product

to find the angle theta

n1 n2

First find the angle between two normal to the planes

cos n1 n2 n1 n2

Find the common perpendicular

1*.cross product uxv is perpendicular to both u and v.

1 2 3

1 2 3

i j k

u u u

v v v

u v

( 2 3 2 3) ( 1 3 3 1) ( 1 2 2 1)u v v u u v u v u v u v i j k

For example: find the common perpendicular to u=(1,2,3) and v=(7,8,9)

1 2 3

7 8 9

i j k

u v (2 9 3 8) (1 9 3 7) (1 8 2 7) i j k

6 12 6 i j k

ˆ ˆ ˆ

x y z

x y z

u u u

v v v

x y z

u v

Find the common perpendicular vector

1. Technique introduced in the textbook.

the common perpendicular vector of l

m

n

and

p

q

r

ismr qn

np lr

lq mp

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 vector1 3 5

0 2 2

i j k

is perpendicular to both AB and AC

Therefore -4i+2j-2k is one of the normal to the plane

4 2 2 i j k

,equation is -4x+2y-2z=-2

5. If a line L has parametric equations

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

3 1, 2 4, 3x t y t z t

Line ALine B

Intersect

Parallel (- identical?)

Skew

Line APoint B

B is on the line AFind theperp. distanceof the point from the line

Line APlane B

Line is in the plane B

A is parallel to B

Line intersects the planeat one point

Point APlane B

Find the perp. distanceof the point from the plane

A is in plane

Angle betweenlines

Plane APlane B

There is a line ofIntersection m

Parallel (- identical?)

Angle betweenplanes

Given the two vector equations of the lines, you can examine thethree simultaneous equations:

1 1 1 1

2 2 2 2

3 3 3 3

a p t b q s

a p t b q s

a p t b q s

If two equations yield a specificpair of values (t,s) and the third equationis 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.

t

s

r a p

r b q

θ 1 .angle cos

pq

p q The angle between two lines is easily found using the dot product.

p

q

To examine whether a point is on a line, simply find a value of theParameter t which satisfies the equation for each coordinate:

1 1 1

2 2 2

3 3 3

a p t m

a p t m

a p t m

1

2

3

t

m

m

m

r a p

m

If there is no value of t which makesthese equations all true, then the pointm 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

p̂

M

A d

X

Line APlane B

Given a plane: . .r n an

and a line: t r b p

… the line is parallel to the plane if: . pn 0

Moreover it is in the plane, also, if:

1 2 3

1 1 1 2 2 2 3 3 3( ) ( ) ( )

n x n y n z d

n b tp n b tp n b tp d

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, . pn 0

Point APlane B

Given a plane: . .r n an

and a point:

1

2

3

m

m

m

m

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 coordinatesof m into the equation of the plane.

To find the distance of the point from the plane:n

A (a,b,c)

px qy rz k d

mˆ.d MA n

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The simplest way is to find a unitnormal vector and then calculate:

Plane APlane B

Given a plane: 1 1 1. .r n a n

and a plane:

The planes may be parallel or identical or …?

2 2 2. .r n a n

Intersect in a line

n1

n2

180 cos 1 2 1 2n n n n

This diagram shows how to find the angle between the planes.

Plane APlane B

Given a plane: 1 1 1. .r n a n

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

2 2 2. .r n a n

Finding a common perpendicular

Read section 13.4 p. 182-3 carefully. The method described is actually the methodto find the cross product.

Find a vector perpendicular to each pair of vectors:

1 7

a) 3 , 2

1 2

3 1

b) 0 , 2

5 3

Note that this method is a good wayto find the equation of a plane throughthree points A, B, C:

AB BC n����������������������������