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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