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**Unformatted text preview: **A unit vector is a vector of length 1 REMARK: If u is a unit vector, then | u | = 1 therefore | u | 2 = 1, and so u u = 1 All the standard basis vectors in R n are unit vectors Sec 5.2.p1 Saturday, February 06, 2010 11:23 PM Section5dot2Part1 Page 1 A unit vector is a vector of length 1 REMARK: If u is a unit vector, then | u | = 1 therefore | u | 2 = 1, and so u u = 1 All the standard basis vectors in R n are unit vectors For any non-zero vector v we can form a unit vector u parallel to v : u = v / | v | u u = ( v / | v | ) ( v / | v | ) = ( 1 / | v | 2 ) v v = ( 1 / | v | 2 ) | v | 2 = 1 So | u | = 1 5.2.p1.1 Saturday, February 06, 2010 11:23 PM Section5dot2Part1 Page 2 A unit vector is a vector of length 1 REMARK: If u is a unit vector, then | u | = 1 therefore | u | 2 = 1, and so u u = 1 All the standard basis vectors in R n are unit vectors For any non-zero vector v we can form a unit vector u parallel to v : u = v / | v | u u = ( v / | v | ) ( v / | v | ) = ( 1 / | v | 2 ) v v = ( 1 / | v | 2 ) | v | 2 = 1 So | u | = 1 | v | 2 = 5 2 + 3 2 + 1 2 = 25 + 9 + 1 = 35 so | v | = 35 and so v is not a unit vector Normalization: v 1 5 3 1 u = = ( 5, 3, 1 ) = ( , , ) | v | 35 35 35 35 5.2.p1.2 Saturday, February 06, 2010 11:23 PM Section5dot2Part1 Page 3 A unit vector is a vector of length 1 REMARK: If u is a unit vector, then | u | = 1 therefore | u | 2 = 1, and so u u = 1 All the standard basis vectors in R n are unit vectors For any non-zero vector v we can form a unit vector u parallel to v : u = v / | v | u u = ( v / | v | ) ( v / | v | ) = ( 1 / | v | 2 ) v v = ( 1 / | v | 2 ) | v | 2 = 1 So | u | = 1 | v | 2 = 5 2 + 3 2 + 1 2 = 25 + 9 + 1 = 35 so | v | = 35 and so v is not a unit vector Normalization: v 1 5 3 1 u = = ( 5, 3, 1 ) = ( , , ) | v | 35 35 35 35 A set of vectors { v 1 , v 2 , . . . , v k } in a vector space V is orthonormal if a) every vector v i is a unit vector ( v i v i = 1 for all i ) b) the vectors are mutually (pairwise) orthogonal ( v i v j = 0 for all i j ) 5.2.p1.3 (*) Saturday, February 06, 2010 11:23 PM Section5dot2Part1 Page 4 Unit vectors: e 1 e 1 = ( 1, 0, 0 ) ( 1, 0, 0 ) = 1 2 + 0 2 + 0 2 = 1 e 2 e 2 = ( 0, 1, 0 ) ( 0, 1, 0 ) = 0 2 + 1 2 + 0 2 = 1 e 3 e 3 = ( 0, 0, 1 ) ( 0, 0, 1 ) = 0 2 + 0 2 + 1 2 = 1 5.2.p1.4 Sunday, February 07, 2010 12:05 AM Section5dot2Part1 Page 5 Unit vectors: e 1 e 1 = ( 1, 0, 0 ) ( 1, 0, 0 ) = 1 2 + 0...

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