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