FCM - Home      Length of a Vector and the Dot Product

  Here we learn about the length of a vector and the angle between two vectors. These are computed by means of two operations on vectors, namely the "norm" and the "dot product". Below we introduce these operations and we describe their properties.

 
Definition   The length or norm of a vector  x (, ... ,)  is given by
 

Thus the norm of a vector is the length of any representing arrow. It has the following properties

Proposition   Properties of the norm operation:
  1. For every  x in   and    in ,   |x| || |x|.    
  2. If  x 0,  the vector  u x/|x|  has length 1 and the same direction as  x;  we call u  the unit vector in the direction of  x.   
  3. All  x,y  in    satisfy the triangle inequality,  |x + y| |x| + |y|
  4. Non degenerate: x in satisfies  x 0   if and only if  x 0.
 

   A more versatile tool than the norm operation is the dot product. It is an operation on two vectors which results in a number. The dot product also allows us to compute the length of vector. In addition it gives the cosine of the angle between two vectors.

Definition   The dot product of two vectors

x ( , ... , )    and    y ( , ... , )

in   is the number

x y + ... + .    

For effective computations with dot products the following rules are frequently helpful.

Proposition      Rules for computing with dot products
For any vectors  a,b,x,y  in    and any scalar    in  , the following hold:
  • bilinearity 
    (a + b) y a y + b y
    a (x + y) a x + a y
    (a) y (a y) a (y)
  • Symmetry:      a x x a
  • Positive definiteness      x x 0, for all  x  in  .
  • Non-degeneracy:      x x 0  if and only if  x 0.

To compute dot products efficiently in more complicated situations, the following formulae can be helpful.

   This is all nice and well. But what is the dot product actually good for? – Its geometrical properties provide answers to this question.

Proposition       geometric properties of the dot product
For any vectors  x  and  y  in  , the following hold:
  • x x |x|
  • x y 0  if and only if  x  is perpendicular to  y.
  • x y |x| |y| cos The angle between the vectors x and y (x,y),   provided  x  and  y  have positive length.
  • |x y| |x| |y|   This is called the Cauchy-Schwarz inequality.

   Exercises   related to the norm and the dot product.

  Move on to any one of the following applications of the dot product