About This Chapter
Analytic Geometry in 3-Dimensions - Chapter Summary
If it has been a long time since your high school or college geometry course, the lessons in this chapter on analytic geometry with 3D objects can help get you back on course. The video lessons here will get you up to speed on volume and surface area calculations of various three-dimensional shapes. Get some practice working with vectors, lines, and planes as well in the lessons on products and parametric equations.
No matter your skill level or how long it's been since you've worked with these mathematical principles, our expert instructors can guide you back into mental shape with this video lecture series. As you progress through this chapter you will get instruction on topics including:
- Calculating volumes of basic 3D shapes
- Working with vectors: dot products, cross products, triple scalar products, and the right hand rule
- Solving line, plane, and parametric equations
- Determining the volume and surface area of cylinders
- Finding the equations for surfaces of revolution
- Understanding level curves and level surfaces
- Determining spherical and cylindrical coordinates
The videos in this chapter provide top-notch instruction in an engaging way with real-life examples acted out using fun animation and graphics. Check out the lesson transcript to see the equations worked out in their entirety and to review key words and functions. Don't forget to take the lesson quizzes and chapter exam to gauge how well you absorbed the instruction and decide if you need to review certain lessons.
1. How to Calculate the Volumes of Basic Shapes
Squares pegs = square holes. Triangular pegs = triangular holes. But where does a sphere go? In this lesson, review volumes of common shapes while contrasting a sphere and a cylinder - after all, they both go into the circular hole... right?
2. How to Solve Visualizing Geometry Problems
A picture is worth a thousand words, but sometimes drawing that picture can be like doing origami with your eyes closed. Practice translating complex problems into simple, meaningful images in this lesson.
3. The Dot Product of Vectors: Definition & Application
After watching this video lesson, you will be able to find the dot product of vectors both algebraically and geometrically. Learn the difference between the two and what you need in order to calculate them.
4. Cross Product & Right Hand Rule: Definition, Formula & Examples
Multiplying two vectors will sometimes give you another vector, known as a cross product, which has many important applications in the real world. In this lesson, you'll learn how to use the right hand rule to find the cross product while exploring some examples.
5. Triple Scalar Product: Definition, Formula & Example
Vectors and scalars are often discussed in physics. In this lesson, we define a particular multiplication of three vectors called the triple scalar product and use an example to show how it is calculated.
6. Lines & Planes in 3D-Space: Definition, Formula & Examples
In this lesson, we will study how lines and planes function in three-dimensional space, and learn how to calculate a line. We also address definitions, formulas and examples.
7. Parametric Equations in Applied Contexts
Watch this video lesson to learn how you can use parametric equations to help you solve real world problems such as the path of a moving object. Learn how to form parametric equations to model the real world.
8. What Are Cylinders? - Definition, Area & Volume
Watch this video lesson to learn why soda cans are cylinders and the method by which you can discover how much soda can fit inside by finding the volume. You will also learn about finding a cylinder's surface area.
9. How to Find Surface Area of a Cylinder
Watch this video lesson to learn the formula you can use to find the surface area of a cylinder. Learn the measurements that you'll need as well as how to use them.
10. Cylindrical & Spherical Coordinates: Definition, Equations & Examples
In this lesson, we introduce two coordinate systems that are useful alternatives to Cartesian coordinates in three dimensions. Both cylindrical and spherical coordinates use angles to specify the locations of points, a feature they share with 2-D polar coordinates.
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Other chapters within the GRE Math: Study Guide & Test Prep course
- Functions in Precalculus
- Analytical Geometry in Precalculus
- Polynomial Equations in Precalculus
- Logarithms & Trigonometry
- Limits of Sequences & Functions
- Calculating Derivatives
- Curve Sketching in Precalculus
- Differentiable Functions & Min-Max Problems
- Indefinite Integrals in Calculus
- Definite Integrals in Calculus
- Additional Topics in Calculus
- L'Hopital's Rule, Integrals & Series in Calculus
- Partial Derivatives
- Calculus: Min/Max & Integrals
- Algebra: Differential Equations
- Algebra: Matrices & Vectors
- Algebra: Determinants & Transformations
- Algebra: Number Theory & Abstract Algebra
- Additional Topics: Sets
- Additional Topics: Unions & Intersections
- Additional Topics: Graphing & Probability
- Additional Topics: Standard Deviation
- Additional Topics: Topology & Complex Variables
- Additional Topics: Trigonometry
- Additional Topics: Theorems, Analysis & Optimizing
- GRE Math Flashcards