# The Three-Dimensional Coordinate System 11.1

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The Three-Dimensional Coordinate System 11.1
JMerrill, 2010

Solid Analytic Geometry
The Cartesian plane (rectangular coordinate system) is determined by 2 perpendicular number line (x- and y-axis) and their point of intersection (the origin). To identify a point in space, we need a third dimension. The geometry of this three-dimensional model is called solid analytic geometry. The 3-D coordinate system is formed by passing a z-axis perpendicular to both the x- and y-axes at the origin.

Coordinate Planes Notice we draw the x- and y-axes in the opposite direction X = directed distance from yz-plane to some point P Y= directed distance from xz-plane to some point P Z= directed distance from xy-plane to some point P (x,y,z) So, to plot points you go out, over, up/down

Octants The 3-D system can have either a right-handed or a left-handed orientation. We’re only using the right-handed orientation meaning that the octants (quadrants) are numbered by rotating counterclockwise around the positive z-axis. There are 8 octants.

Octants

Plotting Points in Space
Plot the points: (2,-3,3) (-2,6,2) (1,4,0) (2,2,-3) Draw a sideways x, then put a perpendicular line through the origin.

Formulas You can use many of the same formulas that you already know because right triangles are still formed.

The Distance Formula It looks the same in space as it did before except with a third coordinate:

Example Find the distance between (1, 0, 2) and (2, 4,-3)

Midpoint Formula The midpoint formula is
What is the midpoint if you make a 100 on a test and an 80 on a test? So the midpoint is just the average of the x’s, y’s, and z’s.

Midpoint You Do Find the midpoint of the line segment joining (5, -2, 3) and (0, 4, 4)

Equation of a Sphere The equation of a circle is x2 + y2 = r2
If the center is not at the origin, then the equation is (x-h)2 + (y-k)2 = r2 The equation of a sphere whose center is at (h,k,j) with radius r is (x-h)2 + (y-k)2 + (z–j)2= r2

Finding the Equation of a Sphere
Find the standard equation of a sphere with center (2,4,3) and radius 3 (x-h)2 + (y-k)2 + (z–j)2= r2 (x-2)2 + (y-4)2 + (z–3)2 = 32 Does the sphere intersect the plane? Yes. The center of the sphere is 3 units above the y-axis and has a radius of 3. It intersects at (2,4,0).

Finding the Center and Radius of a Sphere
Find the center and radius of the sphere given by x2 + y2 + z2 – 2x + 4y – 6z +8 = 0 This works the same way as it did in 2-D space. In order to find the center, we must put the equation into standard form, which means completing the square.

Finding the Center and Radius of a Sphere
x2 + y2 + z2 – 2x + 4y – 6z +8 = 0 (x-1)2 + (y+2)2 + (z-3)2 = 6 The center is (1,-2,3) and the radius is √6.