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Precalculus – MAT 129. Instructor: Rachel Graham Location: BETTS Rm. 107 Time: 8 – 11:20 a.m. MWF. Chapter Ten. Analytic Geometry in Three Dimensions. 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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## Precalculus – MAT 129

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**Precalculus – MAT 129**Instructor: Rachel Graham Location: BETTS Rm. 107 Time: 8 – 11:20 a.m. MWF**Chapter Ten**Analytic Geometry in Three Dimensions**Ch. 10 Overview**• The Three-Dimensional Coordinate System • Vectors in Space • The Cross Product of Two Vectors • Lines and Planes in Space**10.1 – The 3-D Coordinate System**• The 3-D Coordinate System • The Distance and Midpoint Formulas • The Equation of a Sphere**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**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:**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**Example 1.10.1**Pg. 745 Examples 4 & 5 These are the two ways I want you to know how to do these.**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**10.2 – Vectors in Space**• Vectors in Space • Parallel Vectors**10.2 – Vectors in Space**Standard form: v = v1i + v2j + v3k Component form: v = <v1,v2,v3> See all of the properties in the blue box on page 750.**Example 1.10.2**Write the vector v = 2j – 6k in component form.**Solution Example 1.10.2**<0, 2, -6>**10.2 – Angle Between Two Vectors**If Θ is the angle between two nonzero vectors u and v, then cos Θ = u ∙ v / ||u|| ||v||**Example 2.10.2**Pg. 752 Example 3 Simply following the formulas will be all you need to do.**10.2 – Parallel Vectors**Two vectors are parallel when one is just a multiple of the other.**Example 3.10.2**Pg. 752 Example 4**10.2 – Collinear Points**If two line segments are connected by a point and are parallel you can conclude that they are collinear points.**Example 4.10.2**Pg. 753 Example 5**10.3 – The Cross Product of Two Vectors**• The Cross Product • Geometric Properties of the Cross Product • The Triple Scalar Product**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.**Example 1.10.3**Pg. 758 Example 1 You want to leave it in i, j, k form.**10.3 – Geom. Properties of the Cross Product**See the blue box on pg 759. note: orthogonal means perpendicular.**Example 2.10.3**Pg. 759 Example 2 This is the kind of thing you will have to do again.**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.**Example 3.10.3**Pg. 761 Example 4 Pay close attention! As you should remember from determinants, these can be tricky.

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