Copyright © 2011 Pearson, Inc. 8.6 Three- Dimensional Cartesian Coordinate System

Copyright © 2011 Pearson, Inc.

8.6Three-

Dimensional Cartesian

Coordinate System

Slide 8.6 - 2 Copyright © 2011 Pearson, Inc.

What you’ll learn about

Three-Dimensional Cartesian Coordinates Distances and Midpoint Formula Equation of a Sphere Planes and Other Surfaces Vectors in Space Lines in Space

… and whyThis is the analytic geometry of our physical world.


The Point P(x,y,z) in Cartesian Space


The Coordinate Planes Divide Space into Eight Octants


Distance Formula (Cartesian Space)

The distance d(P,Q) between the points P(x1, y

1, z

1)

and Q(x2, y

2, z

2) in space is

d(P,Q) = x1 −x2( )2+ y1 −y2( )

2+ z1 −z2( )

2.


Midpoint Formula (Cartesian Space)

The midpoint M of the line segment PQ with endpoints

P(x1, y

1, z

1) and Q(x

2, y

2, z

2) is

M =x1 + x2

2,y1 + y2

2,z1 + z22

⎛

⎝⎜⎞

⎠⎟.


Example Calculating a Distance and Finding a Midpoint

Find the distance between the points P(1,2,3) and Q(4,5,6),

and find the midpoint of the line segment PQ.


Example Calculating a Distance and Finding a Midpoint

d(P,Q) = 1−4( )2+ 2−5( )

2+ 3−6( )

2

=3 3

The midpoint is M =1+ 42

,2 +52

,3+62

⎛

⎝⎜⎞

⎠⎟=

52,72,92

⎛

⎝⎜⎞

⎠⎟.

Find the distance between the points P(1,2,3) and Q(4,5,6),

and find the midpoint of the line segment PQ.


Drawing Lesson


Drawing Lesson (cont’d)


Standard Equation of a Sphere

A point P(x, y, z) is on the sphere with center (h,k,l)

and radius r if and only if

x −h( )2+ y−k( )

2+ z−l( )

2=r2 .


Example Finding the Standard Equation of a Sphere

Find the standard equation of the sphere with

center (1,2,3) and radius 4.


Example Finding the Standard Equation of a Sphere

x −1( )2+ y−2( )

2+ z−3( )

2=16

Find the standard equation of the sphere with

center (1,2,3) and radius 4.


Equation for a Plane in Cartesian Space

Every plane can be written as Ax + By+Cz+ D=0,where A, B, and C are not all zero. Conversely, every

first-degree equation in three variables represents aplane in Cartesian space.


The Vector v = <v1,v2,v3>


Vector Relationships in Space

For vectors v = v1,v

2,v

3 and w= w

1,w

2,w

3,

g Equality: v = w if and only if v1=w

1, v

2=w

2, v

3=w

3

g Addition: v + w = v1+w

1, v

2+w

2, v

3+w

3

g Subtraction: v −w = v1−w

1, v

2−w

2, v

3−w

3

g Magnitude: v = v1

2 + v2

2 +v3

2

g Dot Product : v ⋅w =v1w

1+v

2w

2+v

3w

3

g Unit Vector : u =v / v , v≠0, is the unit vector in the direction of v.


Equations for a Line in Space

If l is a line through the point P0(x

0, y

0, z

0) in the

direction of a nonzero vector v = a,b,c , then a

point P(x, y,z) is on l if and only if

g Vector form: r = r0 + tv, where r = x, y,z

and r0 = x0 , y0 ,z0 ; or

g Parametric form: x=x0 + at, y=y0 +bt,

and z=z0 + ct, where t is a real number.


Example Finding Equations for a Line

Using the standard unit vector i, j, and k, write a vector

equation for the line containing the points A(−2,0,3) and

B(4,−1,3), and compare it to the parametric equations for

the line.


Example Finding Equations for a Line

The line is in the direction of

v =AB= 7− −5( ),−2−6,−4−0 = 12,−8,−4 .

Using r0 =OA, the vector equation is

r = r0 + tv

x, y,z = −5,6,0 + t 12,−8,−4

x, y,z = −5+12t,6−8t,−4t

xi + yj + zk = −5+12t( )i + 6−8t( ) j + −4t( )k

The parametric equations are the three component equations:x=−5+12t, y=6−8t, z=−4t

A(−5,6,0) and B(7,−2,−4)


Quick Review

Let P(x, y) and Q(3,2) be points in the xy-plane.

1. Compute the distance between P and Q.

2. Find the midpoint of the line segment PQ.

3. If P is 5 units from Q, describe the position of P.

Let v = 4,5 be a vector in the xy- plane.

4. Find the maginitude of v.5. Find a unit vector in the direction of v.


Quick Review Solutions

Let P(x, y) and Q(3,2) be points in the xy-plane.

1. Compute the distance between P and Q. x −3( )2+ y−2( )

2

2. Find the midpoint of the line segment PQ. x+32

,y+ 22

⎛

⎝⎜⎞

⎠⎟

3. If P is 5 units from Q, describe the position of P. x−3( )2+ y−2( )

2=25

Let v= 4,5 be a vector in the xy- plane.

4. Find the maginitude of v. 41

5. Find a unit vector in the direction of v. 4

41,5

41


Chapter Test

1. Find the vertex, focus, directrix, and focal width of

the parabolay2 =12x.

2. Given x−2( )

2

16+

y+1( )2

7=1. Identify the type of

conic, find the center, vertices, and foci.

3. Given x2 −6x−y−3=0. Identify the conic and

complete the square to write it in standard form.

4. Given 2x2 −3y2 −12x−24y+60 =0. Identify the

conic and complete the square to write it instandard form.


Chapter Test

5. Find the equation in standard form for the ellipse

with center (0,2), semimajor axis = 3, and

one focus at (2,2).

6. Find the equation for the conic in standard form.

x =5+3cost, y=−3+3sint, −2π ≤t≤2π.Use the vectors v= −3,1−2 and w= 3,−4,0 .

7. Compute v−w.8. Write the unit vector in the direction of w.


Chapter Test

9. Write parametric equations for the line through

P( −1,0,3) and Q(3,−2,−4).10. B-Ball Network uses a parabolic microphone to

capture all the sounds from the basketball players

and coaches during each regular season game.If one of its microphones has a parabolic surface

generated by the parabola 18y=x2 , locate the

focus (the electronic receiver) of the parabola.


Chapter Test Solutions

1. Find the vertex, focus, directrix, and focal width of the

parabola y2 =12x.vertex (0,0), focus (3,0), directrix x=−3, focal width:12

2. Given x−2( )

2

16+

y+1( )2

7=1. Identify the type of conic,

find the center, vertices, and foci.Ellipse, center (2,−1), vertices (6,−1) (−2,−1),foci (5,−1) ( −1,−1)

3. Given x2 −6x−y−3=0. Identify the conic and

complete the square to write it in standard form.

parabola (x−3)2 =y+12



4. Given 2x2 −3y2 −12x−24y+60 =0. Identify the

conic and complete the square to write it instandard form.

hyperbola y+ 4( )

2

30−

x−3( )2

45=1

5. Find the equation in standard form for the ellipse

with center (0,2), semimajor axis = 3, and

one focus at (2,2).

x2

9+

y−2( )2

5=1



6. Find the equation for the conic in standard form.

x =5+3cost, y=−3+3sint, −2π ≤t≤2π.

x−5( )2

9+

y+3( )2

9=1

Use the vectors v= −3,1−2 and w= 3,−4,0 .

7. Compute v−w. −6,5,−2

8. Write the unit vector in the direction of w. 3 / 5,−4 / 5,0

9. Write parametric equations for the line through

P( −1,0,3) and Q(3,−2,−4).x=−1+ 4t, y=−2t, z=3−7t



10. B-Ball Network uses a parabolic microphone to

capture all the sounds from the basketball players

and coaches during each regular season game.

If one of its microphones has a parabolic surface

generated by the parabola 18y =x2 , locate the

focus (the electronic receiver) of the parabola.(0,4.5)

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Copyright © 2011 Pearson, Inc. 8.6 Three- Dimensional Cartesian Coordinate System