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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

Precalculus – MAT 129

Instructor: Rachel Graham

Location: BETTS Rm. 107

Time: 8 – 11:20 a.m. MWF

chapter ten

Chapter Ten

Analytic Geometry in Three Dimensions

ch 10 overview
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
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 system5
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
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
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
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
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
10.2 – Vectors in Space
  • Vectors in Space
  • Parallel Vectors
10 2 vectors in space11
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
Example 1.10.2

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

10 2 angle between two vectors
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
Example 2.10.2

Pg. 752 Example 3

Simply following the formulas will be all you need to do.

10 2 parallel vectors
10.2 – Parallel Vectors

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

example 3 10 2
Example 3.10.2

Pg. 752 Example 4

10 2 collinear points
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
Example 4.10.2

Pg. 753 Example 5

10 3 the cross product of two vectors
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
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
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
10.3 – Geom. Properties of the Cross Product

See the blue box on pg 759.

note: orthogonal means perpendicular.

example 2 10 3
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
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
Example 3.10.3

Pg. 761 Example 4

Pay close attention! As you should remember from determinants, these can be tricky.