lhe 11.2 three-dimensional coordinate in space calculus iii berkley high school september 14, 2009
TRANSCRIPT
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LHE 11.2Three-Dimensional Coordinate in Space
Calculus IIIBerkley High SchoolSeptember 14, 2009
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Conventions of Three-Dimensional Space With x, y and z axes
perpendicular to each other in three dimensional space, each (a,b,c) of real numbers corresponds to a unique point in space.
Right-Hand Rule…
, , | , ,a b c a b c
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Examples
Graph points
A:(0,5,0),
B:(5,4,6),
C:(1,-1,3)
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Equations in R3
What does z=1 look like?
{(x,y,1)|x,y are R} A plane of height 1
above the xy plane
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Equations in R3
What does y=1 look like?
{(x,1,z)|x,z are R} A plane of distance
1 unit right the xz plane
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Equations in R3
What does x=2 look like?
{(2,y,z)|y,z are R} A plane parallel to
the yz plane and two units in the positive x direction
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Equations in R3
What does x=y look like?
{(x,x,z)|x,z are R} A vertical plane that
crosses through the xy plane through the line x=y
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Equations in R3
xyz=0 {(x,y,z)|x=0 or y=0
or z=0} yz plane union xz
plane union xy plane
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Distance between a point and the origin Find the distance
between the origin the point (1,2,3).
Find the distance between the origin and the point (x, y, z).
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Distance formula from origin to any point in R3
2 2 2d x y z
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Distance between any two points in R3
Find the distance between the given one point A: (x1, y1, z1) and point B: (x2, y2, z2).
If we translate A to the origin then adjust B accordingly, we can use the earlier formula.
2 2 2
2 1 2 1 2 1d x x y y z z
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Defining a sphere
Definition of a sphere centered at the origin: all points equidistant from particular point (center).
2 2 2
2 2 2 2
( , , ) |
( , , ) |
x y z x y z r
x y z x y z r
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Defining a sphere
Definition of a sphere centered at the (a,b,c): all points equidistant from particular point (center).
2 2 2 2x a y b z c r
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What shape is this?
A sphere centered at (2,-1,0) with radius 5^.5
2 2 2
2 2 2
2 2 2
2 2 2
4 2
4 2 0
4 4 2 1 4 1
2 1 5
x y z x y
x x y y z
x x y y z
x y z
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Midpoint formula for R3
1 2 1 2 1 2, ,2 2 2
x x y y z zM
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Vectors in Component Notation
1 1 1, ,x y z
2 1 2 1 2 1, ,x x y y z z
2 2 2, ,x y z
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Directed Line Segment vs. Vector
1 1 1, ,x y z
2 1 2 1 2 1, ,x x y y z z
2 2 2, ,x y z
0,0,0
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Special Unit Vectors
1,0,0
0,1,0
0,0,1
i
j
k
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Are two vectors parallel?
Vectors and are parallel if
there exists a such that
Example
3, 4, 1 , 12, 16,4
When 4,
u v
c
cu v
u v
c cu v
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Are three points collinear?
The points , and form two
vectors: and .
The two vectors are parallel if and only if
the three points are collinear.
Example
1, 2,3 , 2,1,0 , 4,7, 6
2 1,1 2,0 3 1,3, 3
4
P Q R
PQ PR
P Q R
PQ
PR
����������������������������
��������������
��������������1,7 2, 6 3 3,9, 9
3
, , and collinear
PQ PR
P Q R
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Assignment
Section 11.2, 1-67, odd, x61.