1. Inner Products, Length, and Orthogonality (11/30/05) If v and w are vectors in R n, then their inner (or dot) product is: v  w = v T w That is, you multiply the corresponding entries and add up, so the result is a scalar. For example, (2,1,5)  (4,-3,1) = 10

2. Length (or Norm) The length or norm of a vector v is ||v|| =  (v  v) For example, ||(2,5,-1)|| =  30 A unit vector is a vector whose length is 1. Note that for any vector v, the vector v / ||v|| is a unit vector (it has been “normalized”).

3. Distance between two vectors The distance between two vectors v and w is just ||v – w|| For example, the distance between (2,5,-1) and (1,-3,-4) is ||(1,8,3)|| =  74.

4. Orthogonality Two vectors v and w are said to be orthogonal if v  w = 0. Orthogonality generalizes the idea of perpendicularity. A vector v is orthogonal to a subspace W if v is orthogonal to every vector in W. The set of all such vectors is called the orthogonal complement of W.

5. Assignment for Friday Read Section 6.1 carefully. Read these summarizing slides. On pages 382-3, do Exercises 1-19 odd and 27.