Sec 13.5 Equations of Lines and Planes

A line L in three dimensional space is determined when we know a point on L and the direction of L.

:equation vector get the eaddition w for vector

by the and ,scalar somefor ,, Then

.,,

ectorposition vwith on point arbitrary an be ,, and

, ofdirection in the vector a be ,, ,

ector position vwith on point a be ,Let

0

0,000

0,00

LawTriangle

tctbtta tPP

zyx

LzyxP

Lcba ,zyx

LzyxP0

v

r

vr

, , ,,

r r

000

0

tcztbytaxzyx

t

v

Parametric and Symmetric Equations of the line L:

Equating the components of the vector equation, we obtain the parametric equations:

where t is a real number.

Eliminating the parameter t from the parametric equations, we obtain the symmetric Equations of L:

Note: The numbers a, b, and c are called the direction numbers of L.

c

zz

b

yy

a

xx 000

, , 000 ctzzbtyyatxx

Planes

A plane Π in space is determined when we know a point in the plane and a vector perpendicular (orthogonal) to the plane.

0

0

00

0,000

0,00

or

0 -

: plane theof theHence,

. toorthogoanl is and plane in the lies Then

.,,

ectorposition vwith on point arbitrary an be ,, and

, plane thevector to a be ,, ,

ector position vwith on point a be ,Let

r nr n

rr n

n r -r

r

nr

Π

ΠPP

zyx

ΠzyxP

Πcba ,zyx

ΠzyxP0

equation vector

normal

The scalar equation of the plane Π

Using the component forms of the vectors in the vector equation, we obtain the scalar equation:

Note: The equation ax + by + cz + d = 0 is called the linear

equation of the plane.

). (- where

, 0 or

0 or

0 , , ,,

000

000

000

czbyaxd

d czbyax

zzcyybxxa

zzyyxxcba

The distance from a point P to a plane Π

A Distance Formula:

222

111

111

bygiven is 0 :

plane a topoint a from distance The

cba

d czbyax D

d czbyaxΠ

zyxPD

Sec 13.6 Cylinders and Quadratic Surfaces

A cylinder is a surface that consists of all lines (called

rulings) that are parallel to a given line and pass through

a given plane curve C.

The curve C is called the generating curve of the cylinder.

A quadric surface is the graph of a second degree equation

in three variables, x, y, and z. By translation and rotation,

these equations can be reduced to one of the two standard

forms:

0 or

0 22

222

IzByAx

JCzByAx

For graphs of quadric surfaces

See Table 1 on page 844.