Subject:
Engineering, Higher Education, Mathematics, Geometry, Trigonometry
Material Type:
Module
Provider:
Ohio Open Ed Collaborative
Tags:
Area of a Triangle, Component of Projection, Decompostion of a Vector, Dot Products, Force, Normal Vector, Projection, Tmm0022, Unit Vector, Work
License:
Creative Commons Attribution-NonCommercial 4.0
Language:
English
Media Formats:
Downloadable docs

Dot Product and Projection

Module Overview

Vectors - dot product, projection, decomposition of a vector

TMM 002 PRECALCULUS (Revised March 21, 2017)

AdditionalOptional Learning Outcomes:

2. Geometry: The successful Precalculus student can:

2e. Interpret the result of vector computations geometrically and within the confines of a particular applied context (e.g., forces).
Sample Tasks:

  • The student can define vectors, their arithmetic, their representation, and interpretations.
  • The student can decompose vectors into normal and parallel components.
  • The student can interpret the result of a vector computation as a change in location in the plane or as the net force acting on an object.

Section 1: Dot Product and Projection

In this section we will learn about the dot product of two vectors which is quite useful in higher-dimensional geometry, physics, statistics, etc. For example, dot product makes it very easy to find the shortest distance between a line and a point not on the line.

Learning Objectives:

  • Definition of dot product
  • Properties of dot product
  • Geometric Interpretation of Dot Product
  • Orthogonality
  • Projection of a vector onto another vector
  • Component of projection
  • Decomposing a given vector as the sum of two orthogonal vectors
  • Applications of Dot Product
    • shortest distance between a point and a line.
    • Force and work done
    • Cauchy-Schwartz Inequality