Precalculus

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Precalculus MAT 129

• Instructor Rachel Graham
• Location BETTS Rm. 107
• Time 8 1120 a.m. MWF

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Chapter Ten

• Analytic Geometry in Three Dimensions

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Ch. 10 Overview

• The Three-Dimensional Coordinate System
• Vectors in Space
• The Cross Product of Two Vectors
• Lines and Planes in Space

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10.1 The 3-D Coordinate System

• The 3-D Coordinate System
• The Distance and Midpoint Formulas
• The Equation of a Sphere

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10.1 The 3-D Coordinate System

• This text book uses a right-handed system
  approach.
• Figure 10.1 on pg. 742 shows a diagram of this
  orientation.
• Note the three planes xy, xz, and zy

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10.1 Distance and Midpoint

• The distance between the points (x1, y1, z1) and
  (x2, y2, z2) is given by the formula
• d
• The Midpoint formula is given by

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10.1 The Equation of a Sphere

• The standard equation of a sphere with center
  (h,k,j) and radius r is given by
• (x h)2 (y k)2 (z j)2 r2

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Example 1.10.1

• Pg. 745 Examples 4 5
• These are the two ways I want you to know how to
  do these.

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Activities (746)

• 1. Find the standard equation of a sphere with
  center (-6, -4, 7) and intersecting the y-axis at
  (0, 3, 0).
• 2. Find the center and radius of the sphere given
  by .
• x2 y2 z2 - 6x 12y 10z 52 0

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10.2 Vectors in Space

• Vectors in Space
• Parallel Vectors

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10.2 Vectors in Space

• Standard form v v1i v2j v3k
• Component form v ltv1,v2,v3gt
• See all of the properties in the blue box on page
  750.

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Example 1.10.2

• Write the vector v 2j 6k in component form.

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Solution Example 1.10.2

• lt0, 2, -6gt

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10.2 Angle Between Two Vectors

• If T is the angle between two nonzero vectors u
  and v, then
• cos T u v / u v

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Example 2.10.2

• Pg. 752 Example 3
• Simply following the formulas will be all you
  need to do.

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10.2 Parallel Vectors

• Two vectors are parallel when one is just a
  multiple of the other.

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Example 3.10.2

• Pg. 752 Example 4

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10.2 Collinear Points

• If two line segments are connected by a point and
  are parallel you can conclude that they are
  collinear points.

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Example 4.10.2

• Pg. 753 Example 5

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10.3 The Cross Product of Two Vectors

• The Cross Product
• Geometric Properties of the Cross Product
• The Triple Scalar Product

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10.3 The Cross Product

• To find the cross product of two vectors you do
  the same steps as if you were finding the
  determinant of a matrix.
• Note the algebraic properties of cross products
  in the blue box on pg. 757.

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Example 1.10.3

• Pg. 758 Example 1
• You want to leave it in i, j, k form.

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10.3 Geom. Properties of the Cross Product

• See the blue box on pg 759.
• note orthogonal means perpendicular.

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Example 2.10.3

• Pg. 759 Example 2
• This is the kind of thing you will have to do
  again.

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10.3 The Triple Scalar Product

• When we move up a dimension we get to a triple
  scalar product which is a combination of the
  stuff that we have learned so far.
• See the blue boxes on pg. 761.

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Example 3.10.3

• Pg. 761 Example 4
• Pay close attention! As you should remember from
  determinants, these can be tricky.