On this page, we discuss the norm or length of a vector, how to use it to find the distance between two points, and the so-called dot product of two vectors, and its relationship with the angle between the vectors.


The basic and advanced learning objectives listed below are meant to give you an idea of the material you should learn about this section. These are mainly intended to be used in a course which uses an Active Learning approach, where students are required to “read ahead” before each class - but can equally be used in a more traditional course setting.

Unless your teacher gives you specific instructions, it is up to you to decide how much of the listed resources you need to read or watch - you probably do not need to go through all of it. You might also want to look at the General Study Tips & Tricks page for some recommendations on how to effectively study with a math textbook and videos.

Basic learning objectives

These are the tasks you should be able to perform with reasonable fluency when you arrive at your next class meeting. Important new vocabulary words are indicated in italics.

  • Compute the norm of a vector in \(\mathbb{R}^n\).

  • Determine whether a given vector in \(\mathbb{R}^n\) is a unit vector.

  • Normalize a nonzero vector in \(\mathbb{R}^n\).

  • Determine the distance between two points in \(\mathbb{R}^n\).

  • Compute the dot product of two vectors in \(\mathbb{R}^n\).

  • Compute the cosine of the angle between two nonzero vectors in \(\mathbb{R}^n\).

Advanced learning objectives

In addition to mastering the basic objectives, here are the tasks you should be able to perform after class, with practice:

  • Compute the angle between two nonzero vectors in \(\mathbb{R}^n\).

  • Solve problems involving norms and dot products.

  • Prove basic properties pertaining to norms and dot products.

To prepare for class

  • Watch this short video which explains how to find the vector coordinates of a vector between two points, and how to find its length (or “norm” or “magnitude”):

  • Watch this short video which explains how to normalize a vector which does not yet have length 1:

  • Watch just the first 4 minutes of this video (again by 3Blue1Brown) which introduces the dot product (the later part of the video explains a deeper connection of this with the concepts of “duality” and “linear transformations”, which you might want to come back to much later, possibly at the very end of your course - but you can ignore this for now):

  • Watch this video (by PatrickJMT) which gives an example on how to find the angle between two vectors, using their dot product:


Gabriel Indurskis Avatar Gabriel Indurskis






Please click here if you find a mistake or broken link/video, or if you have any other suggestions to improve this page!