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

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.

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