# Chapter 5: Vectors

**At Grade**Created by: CK-12

## Introduction

We use vectors to describe how different forces work together. They can be thought of as measures of "shared contribution" to a specific goal, usually a speed or distance traveled.

For instance, consider a boat traveling across a stream. The boat would be pushed straight across by the motor, but the current of the stream would be pushing it sideways to that motion, so the result would be a diagonal direction. In this example, the push of the motor and the push of the current would each be a *vector*, resulting in the diagonal travel direction.

Vector calculations can be used in a vast number of different situations, to combine or separate different contributions to a total.

## Chapter Outline

- 5.1. Two-Dimensional Vectors
- 5.2. Positions and Midpoints in Two Dimensions
- 5.3. Three-Dimensional Positions
- 5.4. Dot Products
- 5.5. Scalar Projections
- 5.6. Cross Products
- 5.7. Planes in Space
- 5.8. Distance Between a Point and a Plane
- 5.9. Vector Direction
- 5.10. Vector Equation of a Line
- 5.11. Vector Analysis Applications

### Chapter Summary

## Summary

This chapter is a reasonably in-depth exploration of the uses of vectors. The information begins with a study of 2-dimensional vectors and the calculation of midpoint and location in a 2-dimensional plane, and progresses through similar topics in 3-dimensions.

Students are later introduced to dot-products and cross-products and the uses for each, and the associated applications of scalar and vector projections.

The last few lessons cover the calculation of the distance of a plane from the origin, and the vector equation of a line, before summing up with examples of vector analysis in real-world situations.