notes6-1 - SECTION 6.1 INNER PRODUCT LENGTH ORTHOGONALITY...

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SECTION 6.1 INNER PRODUCT, LENGTH, ORTHOGONALITY We return (temporarily) to R n . The inner product or dot product of two vectors u = u 1 u 2 . . . u n and v = v 1 v 2 . . . v n in R n is u 1 v 1 + u 2 v 2 + · · · + u n v n . The inner product has natural nice algebraic properties listed on page 376. It gives rise to the length or norm of a vector u in R n given by || u || = u · u . This agrees (at least in R 2 ) with our understanding of the length of the line segment from the origin to the tip of u . The distance between vectors u and v in R n is dist( u , v ) = || u - v || . Again, this agrees with the geometry, at least in R 2 . THE GEOMETRY IN R 2 . Let’s do a little checking of the algebra, in particular, let’s see what happens to all this stuff when we multiply by a scalar. What about ( c u ) · v ? What about || c u || ? A unit vector is a vector whose length is 1. If we divide a nonzero vector by its length, we’ll get a unit vector in the same direction. Let’s check this.
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