On this page, we discuss the cross product of two 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 cross product of two vectors in \(\mathbb{R}^3\).
  • Know the geometric relationship between \(\vec{u}\times \vec{v}\) to \(\vec{u}\) and \(\vec{v}\).
  • Know the geometric meaning of the magnitude of \(\vec{u}\times \vec{v}\).
  • Know the properties of the cross product.
  • Compute the scalar triple product of three vectors in \(\mathbb{R}^3\).
  • Know the geometric interpretation of the scalar triple product.
  • Use the scalar triple product to determine whether three given vectors in 3-space are coplanar.

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 areas of triangles and parallelograms determined by two vectors or three points in \(\mathbb{R}^2\) or \(\mathbb{R}^3\).
  • Use the cross-product to quickly find the normal equation of a plane passing through three given points.
  • Solve problems involving points, vectors, lines, and planes, in \(\mathbb{R}^3\).

To prepare for class

  • Watch this video about the definition of the cross-product and the right-hand rule:

  • Watch this video about how to use the cross-product to find a normal vector (and therefore a normal equation) for a plane through 3 given points (note that he uses the so-called “determinant method” to calculate the cross-product - you can ignore this and simply use the “usual” method instead. Otherwise, you could read a bit ahead and read about \(2\times 2\) and \(3\times 3\) determinants on this page:

  • Watch this video about how to use the cross-product to find the area of a triangle in \(\mathbb{R}^3\):

After class

  • Watch this video which compares the dot and cross products of two vectors, and their relationship to the angle between the vectors:


Gabriel Indurskis Avatar Gabriel Indurskis






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