Mathematics Chap 12
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Transcript of Mathematics Chap 12
![Page 1: Mathematics Chap 12](https://reader035.fdocuments.in/reader035/viewer/2022062310/56d6c0081a28ab301698ad44/html5/thumbnails/1.jpg)
THREE DIMENSIONAL GEOMETRY
INTRODUCTION
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TWO-DIMENSIONAL (2-D) COORDINATE SYSTEMS
To locate a point in a plane, two numbers are necessary.
We know that any point in the plane can be represented as an ordered pair (a, b) of real numbers—where a is the x-coordinate and b is the y-coordinate.
For this reason, a plane is called two-dimension
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THREE-DIMENSIONAL (3-D) COORDINATE SYSTEMS To locate a point in space,
three numbers are required.
We represent any point in space by an ordered triple (a, b, c) of real numbers
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THREE-DIMENSIONAL (3-D) COORDINATE SYSTEMS In order to represent points in
space, we first choose:
A fixed point O (the origin)
Three directed lines through O that are perpendicular to each other
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COORDINATE AXES The three lines are called the
coordinate axes. They are labeled:
x-axisy-axisz-axis
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COORDINATE AXESWe draw the orientation of the axes as shown.
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COORDINATE PLANES The three coordinate axes
determine the three coordinate planes.
i. The xy-plane contains the x- and y-axes.
ii. The yz-plane contains the y- and z-axes.
iii. The xz-plane contains the x- and z-axes.
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OCTANTS
These three coordinate planes divide space into eight parts, called octants.
The first octant, in the foreground, is determined by the positive axes
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3-D COORDINATE SYSTEMS- EXAMPLE
1) Look at any bottom corner of a room The wall on your left is in the xz-plane.
2) The wall on your right is in the yz-plane.
3) The floor is in the xy-plane.4) and call the corner the origin.
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3-D COORDINATE SYSTEMS
Now, if P is any point in space
We represent the point P by the ordered triple of real numbers (a, b, c).
We call a, b, and c the coordinates of P.
a is the x-coordinate.b is the y-coordinate.c is the z-coordinate.
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DISTANCE FORMULA IN THREE DIMENSIONS
The distance |P1P2| between the points P1(x1,y1, z1) and P2(x2, y2, z2) is:
2 2 21 2 2 1 2 1 2 1( ) ( ) ( )PP x x y y z z
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PROOF OF DISTANCE FORMULA
To see why this formula is true, we construct a rectangular box as shown, where:
P1 and P2 are opposite vertices.
The faces of the box are parallel to the coordinate planes
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PROOF
If A(x2, y1, z1) and B(x2, y2, z1) are the vertices of the box, then
|P1A| = |x2 – x1|
|AB| = |y2 – y1|
|BP2| = |z2 – z1|
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PROOF
Triangles P1BP2 and P1AB are right-angled.
So, two applications of the Pythagorean Theorem give:
|P1P2|2 = |P1B|2 + |BP2|2
|P1B|2 = |P1A|2 + |AB|2
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PROOF
I. Combining those equations, we get:
II. |P1P2|2 = |P1A|2 + |AB|2 + |BP2|2
III. = |x2 – x1|2 + |y2 – y1|2 + |z2 – z1|2
IV. = (x2 – x1)2 + (y2 – y1)2 + (z2 – z1)2
Therefore,2 2 2
1 2 2 1 2 1 2 1( ) ( ) ( )PP x x y y z z
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EXAMPLE OF DISTANCE FORMULA The distance from the point P(2, –1,
7) to the point Q(1, –3, 5) is:
2 2 2(1 2) ( 3 1) (5 7)
1 4 43
PQ
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SECTION FORMULA
INTERNAL DIVISION EXTERNAL DIVISION
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MID POINT FORMULA
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SOLVED EXAMPLE OF MID POINT FORMULA
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CENTROID OF A TRIANGLE
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The Centroid of a Triangle is usually represented by G
Therefore
G=(x1+x2+x3/3, y1+y2+y3/3,z1+z2+z3/3)
FORMULA FOR CENTROID OF A TRIANGLE