MAT 1236Calculus III

Section 12.5 Part II

Equations of Line and Planes

http://myhome.spu.edu/lauw

HW…

WebAssign 12.5 Part II • 28 problems, start it ASAP

Be sure to think about the solutions method carefully.

There can be a lot of variations • It is not practical to memorize all the cases.

You need to understand the principles and practice solving problems.

Hints for the last problem is in the HO.

Quiz

Quiz: 12.3, 12.4, 12.5 I

Preview

Equations of Planes• Vector Equations

• Scalar Equations

Similar development as in lines

Vector Equations of Planes

Ingredients• A (fixed) point on the plane

• A (fixed) vector n=<a,b,c> orthogonal to the plane

n is called a ________________ For a (general) point on the plane,

________________

0 0 0 0, ,P x y z

, ,P x y z

Scalar Equations

0

0 0 0

0

, , , , , , 0

n r r

a b c x y z x y z

Example 1

Find an equation of the plane through the point P(-5,1,2) with normal vector

n=<3,-5,2>.

Example 1

Find an equation of the plane through the point P(-5,1,2) with normal vector

n=<3,-5,2>.

Can you recover n=<3,-5,2> from the linear equation?

Example 2

Find the equation of the plane through the points A(-1,1,-1), B(1,-1,2), C(4,0,3)

5 7 8

5 7 8 4 0

i j k

x y z

Parallel Planes

Two planes are parallel if their normal vectors are_______.

Parallel Planes

If two planes are not parallel, then they intersect in a straight line and the angle between the two planes is defined as the acute angle between their normal vectors.

Example 3 (a)

Find the angle between the two planes

1

2

: 5

: 3 1

P x y z

P x y z

1.02

Example 3 (b)

Find symmetric equations of the line of intersection.

1

2

: 5

: 3 1

P x y z

P x y z

Example 3 (b) Method I

Find symmetric equations of the line of intersection.

1

2

: 5

: 3 1

P x y z

P x y z

Geometric Meanings

Unless the line is horizontal, there is a point on it with a zero z coordinate.

, ,0x y

Geometric Meanings

Unless the line is horizontal, there is a point on it with a zero z coordinate.

, ,0x y

, ,0x y

0

plane

z

x y

Geometric Meanings

Unless the line is horizontal, there is a point on it with a zero z coordinate.

, ,0x y

, ,0x y

0

plane

z

x y

1

2

: 5

: 3 1

P x y z

P x y z

Example 3 (b) Method II

1

2

: 5

: 3 1

P x y z

P x y z

2n1n

2P1P

1 2n n

LL

Example 4

Find the distance from a point

to the plane

(Reading Assignment: read this from the text)

1 1 1 1, ,P x y z

0ax by cz d

a

a bcomp b

a