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

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P Q R

PQ PR

P Q R

PQ

PR

����������������������������

��������������

��������������1,7 2, 6 3 3,9, 9

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, , and collinear

PQ PR

P Q R

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Assignment

Section 11.2, 1-67, odd, x61.