Embed Size (px)
Citation preview
1.3 Lines and Planes
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
The vector form is
Lines in R2 and R3
],,[ v cba
vpx t
The parametric equations of a line in the plane is btyy
atxx
1
1
ctzz
btyy
atxx
1
1
1
Find vector and parametric 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
Example
To determine a line L, we need a point P(x1,y1) on Land a normal vector that is perpendicular to L.
The normal form of the equation of a line in the plane is
The general form of the equation is
Lines in the Plane
],[ n ba
0pxn )(
0 cbyax
To determine a plane P, we need a point P(x1,y1,z1) on Pand a normal vector that is orthogonal to P.
The normal form of the equation of a plane is
The general form of the equation of a plane is
Equation of a Plane
],,[ n cba
0pxn )(
0 dczbyax
Two planes in space with normal vectors n1 and n2 are either parallel or intersect in a line. They are parallel iff their normal vectors are. They are perpendicular iff their normal vectors are.
To determine a plane P, we need a point P(x1,y1,z1) on Pand two direction vectors and that are parallel to P.
The vector form of the equation of a plane is
The parametric form of the equation of a plane is
Equation of a Plane
],,[ 321 u uuu
vupx ts
],,[ 321 v vvv
331
221
111
tvsuzz
tvsuyy
tvsuxx
Find parametric and general forms of the 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
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.
),( PQprojPQd D 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
