Post on 23-Dec-2015
transcript
Vectors: planes
2.An example in (1, 2) .Find the cartesian equation of the line through perpendicular to the non-zero vector
l
m
(1, 2)
Let be the perpendicular to the line
and be the position vector of . Let be
the position of any other point on the line.
l
m
n
a r
. 0 . .Then , or r a n r n a n
1
2If and we have . .
l x x l ln r
m y y m m
2i.e. l x my l m
The plane3A plane in has the special property that
perpendiculars to it are in the same direction
at every point.
These perpendiculars are called .normals
Let be a vector perpendicular to the plane
and let be the position vector of a point in the
plane. Let be the position vector of any other point
on the plane.
n
a
r
). 0 . .
Then is a vector parallel to the plane and, hence, perpendicular
to . Therefore ( or
r a
n r a n r n a n
Normal equation of the plane
. . .
Points of a plane through and perpendicular to the normal
vector have position vectors which satisfy
This is called the of the plane.
A
normal equation
n r r n a n
Cartesian equation of the plane
, . .
.
Writing and the equation
becomes
x p
y q
z r
px q y r z
r n r n a n
a n
So, points of a plane through and perpendicular to the normal
vector (or ) have coordinates which satisfy
, where is a constant determined by the coordinates
A
p
q p q r
r
px q y r z k k
n n i j k
.of This is called the of the plane.A cartesian equation
An example
Intersection of a line and a plane
Does a line lie in a plane?Do Exercise 13A, p.179
Distance from a point to a plane
Do Q6, Q7, pp185-186.
Angle between a line and a plane
Do Q10, p.186
Finding a common perpendicular
If and are non-zero, non-parallel vectors,
then is non-zero and perpendicular to both of them.
l p
m q
n r
mr nq
np lr
lq mp
Remembering this vector:
Plane through three points(1,2,1)
(2, 1, 4) (1,0, 1)
Find the cartesian equation of the plane through ,
and .
A
B C
2 1 1 1 1 0
1 2 3 0 2 2
4 1 5 1 1 2
and are both parallel to the plane.
����������������������������AB AC
3 ( 2) ( 5) ( 2) 4
( 5) 0 1 ( 2) 2 .
1 ( 2) ( 3) 0 2
A vector normal for the plane is
2
1 .
1
A simpler plane normal is
2So the plane has cartesian form .x y z k
(1, 2,1) 2 1.Since the plane passes through , it's equation is x y z
Do Q1-Q5, Q14, pp.185-186
Line of intersection of two planes1 2Find the line of intersection of the planes and . x y z x y z
0 1 2
3 1 3 1, , ,0 .2 2 2 2
Let to obtain the simultaneous equations and , which has
solution , so the line of intersection contains the point
z x y x y
x y
0
1
1
The line of intersection has direction vector perpendicular to both plane normals;
use the formula to obtain as a direction vector.
3 2 0
1 2 1 .
0 1
So the vector equation for the line of intersection is t
r
Note that the angle between two intersecting planes is defined as the angle between their normals.
Do Q12, Q13, p.186 and Q2, p.186