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Unformatted text preview: MA 265 LECTURE NOTES: WEDNESDAY, MARCH 26 Inner Product Spaces Direction in R 2 and R 3 . We have seen that the angle between two vectors u and v in R 2 is given by = arccos u 1 v 1 + u 2 v 2  u  v  where . Similarly, the angle between two vectors u and v in R 2 is given by = arccos u 1 v 1 + u 2 v 2 + u 3 v 3  u  v  where . We use this to make a few of definitions: We say that two nonzero vectors u and v are in the same direction if = 0, i.e., 0 ; orthogonal or perpendicular if = / 2, i.e., 90 ; and in opposite directions if = , i.e., 180 . Inner Products on R 2 and R 3 . There is a simple way to express the formula above. Define the standard inner product or dot product as the scalar u v = ( u 1 v 1 + u 2 v 2 on R 2 ; u 1 v 1 + u 2 v 2 + u 3 v 3 on R 3 . Then we have the following formulas: The length of a vector is  v  = v v . The distance from v to u is  v u  = p ( v u ) ( v u ) = p  u  2 2( u v ) +  v  2 The angle between u and v is = arccos u v  u  v  . Hence it follows that two vectors u and v not necessarily nonzero! are in the same direction if u v =  u   v  ; orthogonal or perpendicular if u v = 0; and in opposite directions if u v = u   v  . Example. Consider the vector x = 3 4 ....
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This note was uploaded on 03/11/2010 for the course MA 261A 0026100 taught by Professor ... during the Spring '10 term at Purdue University Calumet.
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