Unit 6
Vectors in Two and Three Dimensions

Up to this point, you have studied the basics of matrix arithmetic and matrix algebra as it is used to solve systems of linear equations. Historically, this is how the study of matrices began. Parallel to this development, and at about the same time, the theory of vectors in two and three dimensions was being worked out. In the late nineteenth century, these two interconnected lines of work were combined in the theory of n-dimensional linear vector spaces. We introduce this theory in Units 8 and 9.

An n-dimensional vector space is just a generalization of vectors in two or three dimensions. Thus, it is important that you become totally familiar with the content of this unit, and of Unit 7. It is particularly important because two- and three-dimensional constructions can be visualized, while n-dimensional constructions (n > 3) cannot. Hence, you will need to rely on two- or three-dimensional visualizations in working problems in higher numbers of dimensions. This unit offers you the opportunity to practise visualization in cases in which it can be connected to your common-sense ideas of distance and to your kinesthetic awareness of space. Indeed, in many of the problems in this unit, you will find it useful to make a sketch of the problem situation, and then imagine yourself walking around inside it, moving vectors about as if they were physical objects. By involving yourself this way, you will lay a solid foundation for your mathematical intuition.

Objectives

When you have completed this unit, you should be able to

  1. define vectors in two and three dimensions, compute the norm of these vectors, and carry out all the standard operations of vector arithmetic.
  2. define and compute the dot product of two vectors, and determine when two vectors are orthogonal, or more generally, find the angle between any two vectors.
  3. define, compute and describe geometrically the projection of one vector along another vector.
  4. write any vector as a sum of a vector parallel to a given vector v and a vector orthogonal to v.
  5. define and compute the cross product of any two vectors in three dimensions.
  6. state and prove the basic properties of the dot and cross products.
  7. express any vector in terms of the standard unit vectors, and compute the cross product of two vectors in terms of a “formal” determinant using these standard unit vectors.
  8. use your knowledge of vectors to prove various vectorial and geometric identities.