11.5 Lines and Planes in Space

1

1

1

x x at

y y bt

z z ct

1 1 1x x y y z z

a b c

To determine a line L, we need a point P(x1,y1,z1) on Land a direction vector for the line L.

The parametric equations of a line in space is

If a, b and c are nonzero, we can eliminate parameterto obtain the following symmetric equations:

, ,a b cv

Lines in Space

Find sets of parametric equations and symmetric equations of the line

1) through two points

(2,1,1) , (4, 5,3)

5 , 4 2 , 3x t y t z

2) through the point (2,3,1) and is parallel to

Examples

1 1 1( ) ( ) ( ) 0a x x b y y c z z

0ax by cz d

To determine a plane, we need a point P(x1,y1,z1) in the planeand a normal vector that is perpendicular to the plane.

The standard equations of a plane in space is

By regrouping terms, we obtain the general form of the line:

, ,a b cn

Equation of a Plane

Find an equation of the plane passing 1) through the points (3, 1, 2) , (2,1,5) , (1, 2, 2)

6 7 2 10x y z

2) through the points (3,2,1), (3,1,-5) and is perpendicular to

Examples

1 2

1 2

cosn n

n n

1 2n n

To find the line of intersection between two planes, solve the system of two equations with three unknowns. The line of intersection is parallel to

Two planes in space with normal vectors n1 and n2 are eitherparallel or intersect in a line. They are parallel if and only if their normal vectors are. They are perpendicular if and only if their normal vectors are. The angle between two planes is equal to the angle between the normal vectors are given by

Planes in Space

Given two planes with equations 1) Find the angle between two planes.

6 3 5

5 5

x y z

x y z

2) Find the line of intersection of the planes.

Examples

The distance between a plane (with normal vector n) and a point Q (not in the plane) is

where P is any point in the plane.

PQD proj PQ

n

n

n

����������������������������

The distance between a line (with direction vector v) in space and a point Q is

where P is any point on the line.

PQD

v

v

��������������

Distance

Given two planes with equations 1) Find the distance between the point (1,1,0) and the plane P1.

1

2

: 5 9

: 10 2 2 3

P x y z

P x y z

2 , 3 , 2 2x t y t z t 2) Find the distance between the point (1,-2,4) and the line

3) Show that P1 and P2 are parallel.

4) Find the distance between P1 and P2 .

Examples

The distance between a plane with equation

and a point Q(x0,y0,z0) (not in the plane) is

0ax by cz d

The distance between a line in the plane and a point Q(x0,y0) (not on the line) is

0 0 0

2 2 2

ax by cz dD

a b c

0Ax By C

0 0

2 2

Ax By CD

A B

Distance Formulas