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Chapter 12 – Vectors and the Geometry of Space12.1 – Three Dimensional Coordinate Systems
12.1 – Three Dimensional Coordinate Systems
12.1 – Three Dimensional Coordinate Systems
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Coordinate axesThe 3D coordinate plane is used
to represent any point in space.O is the origin.x, y, and z axis are all
perpendicular
12.1 – Three Dimensional Coordinate Systems
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3D Coordinate SystemThe purple plane
is the yz-plane.The pink plane is
the xz-plane.The green plane
is the xy-plane.
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Points in 3D Coordinate SystemsOrdered pairs are in
the form (x, y, z) called an ordered triple.
Here we plotted point P by moving a units along the x-axis, b units along the y-axis, and c units along the z-axis.
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3D Coordinate SystemThe point P(a,b,c) determines a rectangular
box. We drop a perpendicular from P to the xy-plane and get point Q called the projection of P on the xy-plane. Similarly, points P and S are the projections of P on the yz-plane and xz-plane respectively.
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DefinitionThe Cartesian product
is the set of all ordered triples of real numbers and is denoted by .
We have a one-to-one correspondence between the points P in space and the ordered triples (a,b,c) in .
It is called a three dimensional rectangular coordinate system.
( , , ) | , ,x y z x y z
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DefinitionDistance Formula in 3D: the distance
|P1P2|between the points P1(x1,y1,z1) and P2(x2,y2,z2) is
2 2 2
1 2 2 1 2 1 2 1PP x x y y z z
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DefinitionEquation of a Sphere – An equation of a
sphere with center C(h,k,l) and radius r is
In particular, if the center is the origin O, the equation of the sphere is
2 2 22r x h y k z l
2 2 2 2r x y z
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Example 1 – Page 790 #4What are the projections of the
point (2,3,5) on the xy-, yz-, and xz-planes? Draw a rectangular box with the origin and (2,3,5) as opposite vertices and with its faces parallel to the coordinate planes. Label all vertices of the box. Find the length of the diagonal of the box.
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Example 2 – Page 790 # 14Find an equation of the sphere
that passes through the origin and whose center is (1,2,3).
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Example 3 – Page 790 #16Show that the equation
represents a sphere and find its center and radius.2 2 2 8 6 2 17 0x y z x y z
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Example 4 – Page 791 #20Find an equation of a sphere if
one of its diameters has endpoints (2,1,4) and (4,3,10).
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More Examples
The video examples below are from section 12.1 in your textbook. Please watch them on your own time for extra instruction. Each video is about 2 minutes in length. ◦Example 1◦Example 3◦Example 5