8.6 Three-Dimensional Cartesian
Coordinate System
Name: _______________
Objectives: Students will be able to draw three-dimensional
figures and analyze vectors in space.
Feb 27­7:14 AM
Examples Draw a sketch that shows the point.
1.) (2,-3,6)
2.) (-2,3,5)
Feb 27­7:26 AM
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Distance Formula: The distance between the points P(x1,y1,z1)
and Q(x2,y2,z2) is d(P,Q) = √(x1 - x2)2 + (y1 - y2)2 + (z1 - z2)2 .
Example Compute the distance between (2,-1,-8) and (6,-3,4).
Midpoint Formula: The midpoint M of the line segment PQ with
endpoints P(x1,y1,z1) and Q(x2,y2,z2) is
M=
Example Find the midpoint of the segment PQ, where P(2,-1,-8)
and Q(6,-3,4).
Feb 27­7:28 AM
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 Write the equation of the sphere with center (-1,5,8)
and radius √5.
Feb 27­7:45 AM
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Drawing Lesson! Let's take a break from notes and draw. Read
through the drawing lesson on page 688. Use the next two boxes
on the note sheet to practice drawing objects that look threedimensional.
Feb 27­7:49 AM
Feb 27­7:51 AM
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We know already that every LINE can be written in the form
Ax + By + C = 0, where A and B are not both zero.
So...every PLANE can be written as Ax + By + Cz + D = 0, where A,
B and C are not all zero.
Example Sketch the graph of 12x + 15y + 20z = 60.
Feb 27­7:51 AM
We also have vectors in three-dimensional space and all of the
vector rules that we've previously learned just extend to three
dimensions. (See the rules on page 960 of your textbook.)
Examples Let r = <1,2,3>, v = <-1,0,5> and u = <0,-2,7>. Find the
following:
a.) (r v) + (v u)
b.) r
c.) unit vector in the direction of v
Assignment: Page 963: #1-31 odd
Feb 27­7:54 AM
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