Workshop: Inner Product, Length, and Orthogonality

Below are the overview and guided practice materials for our third workshop, on Inner Product, Length, and Orthogonality.


Our last big application of this course -- least-squares methods of solving systems -- requires a set of tools regarding the length of vectors in Rn and the angles at which vectors meet. This workshop will focus on using some basic mechanical calculations to get to a few main concepts that will emerge later. Those tools are the inner product (a way of multiplying two vectors to get a numerical result that contains information about the angle at which the vectors meet), the notion of the length or norm of a vector (which we use to normalize vectors), and the concept of orthogonality (a generalized notion of when two vectors are perpendicular).


For this workshop, there are no screencasts, but please read from page 375 to Theorem 6.2 (including the statement of Theorem 6.2) at the top of page 380 in your textbook. Remember that reading a textbook is more than just "reading". It involves working through the examples on your own, making examples and exercises of your own and working them out, and trying exercises from the set at the end of the section. As you work, you would do well to work through Practice Problems 1--4 on page 382 (solutions are on page 384) as well as some of the exercises listed in the Competencies list for this section.

Note: You do not need to read the material on orthogonal complements or angles in R2 and R3 at the end of the section. However, some of your workshop problems might draw upon this material.

If you need extra examples or a second presentation of this material, try the following videos from Khan Academy:


By class time on Monday April 25, each student should be able to perform the following tasks and respond correctly on a 5-minute quiz:

  • Calculate the inner product of two vectors.
  • State and use the inner product properties listed in Theorem 6.1.
  • Calculate the norm (length) of a vector.
  • Determine whether a vector is or is not a unit vector.
  • Given a nonzero vector, find a unit vector that points in the same direction. (In other words, "normalize" the vector.)
  • Find the distance between two vectors.
  • Determine whether two vectors are or are not orthogonal.


Please complete the following exercises from the Lay textbook, starting on page 382: 2, 4, 6, 8, 10, 14, and 16. Please work all these out by hand and show your work. You may benefit from doing the odd problems near these first, so you can check your answers.

On Monday, April 25:
  • Turn in your work on the exercises above.
  • Be prepared to take a five-question clicker quiz on the competency list above.
  • We will work in groups on some more difficult tasks in which you will put the basic knowledge you acquire to work.