copyright © 2011 pearson, inc. 8.6 three- dimensional cartesian coordinate system
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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.
Slide 8.6 - 3 Copyright © 2011 Pearson, Inc.
The Point P(x,y,z) in Cartesian Space
Slide 8.6 - 4 Copyright © 2011 Pearson, Inc.
The Coordinate Planes Divide Space into Eight Octants
Slide 8.6 - 5 Copyright © 2011 Pearson, Inc.
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.
Slide 8.6 - 6 Copyright © 2011 Pearson, Inc.
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
⎛
⎝⎜⎞
⎠⎟.
Slide 8.6 - 7 Copyright © 2011 Pearson, Inc.
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.
Slide 8.6 - 8 Copyright © 2011 Pearson, Inc.
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.
Slide 8.6 - 9 Copyright © 2011 Pearson, Inc.
Drawing Lesson
Slide 8.6 - 10 Copyright © 2011 Pearson, Inc.
Drawing Lesson (cont’d)
Slide 8.6 - 11 Copyright © 2011 Pearson, Inc.
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 .
Slide 8.6 - 12 Copyright © 2011 Pearson, Inc.
Example Finding the Standard Equation of a Sphere
Find the standard equation of the sphere with
center (1,2,3) and radius 4.
Slide 8.6 - 13 Copyright © 2011 Pearson, Inc.
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.
Slide 8.6 - 14 Copyright © 2011 Pearson, Inc.
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.
Slide 8.6 - 15 Copyright © 2011 Pearson, Inc.
The Vector v = <v1,v2,v3>
Slide 8.6 - 16 Copyright © 2011 Pearson, Inc.
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.
Slide 8.6 - 17 Copyright © 2011 Pearson, Inc.
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.
Slide 8.6 - 18 Copyright © 2011 Pearson, Inc.
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.
Slide 8.6 - 19 Copyright © 2011 Pearson, Inc.
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)
Slide 8.6 - 20 Copyright © 2011 Pearson, Inc.
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.
Slide 8.6 - 21 Copyright © 2011 Pearson, Inc.
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
Slide 8.6 - 22 Copyright © 2011 Pearson, Inc.
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.
Slide 8.6 - 23 Copyright © 2011 Pearson, Inc.
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.
Slide 8.6 - 24 Copyright © 2011 Pearson, Inc.
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.
Slide 8.6 - 25 Copyright © 2011 Pearson, Inc.
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
Slide 8.6 - 26 Copyright © 2011 Pearson, Inc.
Chapter Test Solutions
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
Slide 8.6 - 27 Copyright © 2011 Pearson, Inc.
Chapter Test Solutions
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
Slide 8.6 - 28 Copyright © 2011 Pearson, Inc.
Chapter Test Solutions
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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