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Ch. 10 – 3D Analytic Geometry

10.1 – The 3D Coordinate System

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The Coordinate System Positive x-direction comes “out of the page”

z Positive x-direction comes “out of the page” Positive y-direction goes to the right Positive z-direction goes straight up xy plane y xz plane x yz plane

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The Coordinate System x y z Instead of quadrants, we now have octants, designated in the figure by cubes Octant I has x, y, and z positive (closest to you) Octants II – IV are counterclockwise around the z-axis Octants V-VIII are below Octants I – IV, respectively Octant II Octant I Octant V

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The Coordinate System Plot the following points: (2, 3, 3) (3, -1, 2)

x y z Plot the following points: (2, 3, 3) (3, -1, 2) (1, 4, 0)

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Distance Formula In 2D, distance formula and midpoint formula are…

In 3D, the formulas change to… Ex: Find the distance between (4, 1, 1) and (-3, 2, 0). Then find the midpoint of those two points.

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Spheres In 2D, the general formula for a circle is…

In 3D, the general formula for a sphere is… Ex: Find the center and radius of the sphere with the equation x2 + y2 + z2 – 2x + 4y – 6z + 8 = 0. Complete the square! (h, k) is the center r is the radius (h, k, j) is the center r is the radius Center = (1, -2, 3)

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