13.2-4 Properties of Vector Operations

13.2-4 Properties of Vector Operations - Properties of...

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Unformatted text preview: Properties of Vector Operations Addition and Scalar Multiplication 1. a + b = b + a 3. a + 0 = a 7. (cd)a = c(da) 2. 4. 8. a + (b + c) = (a + b) + c a + (-a) = 0 (c + d)a = ca + da 1a = a 5. c(a + b) = ca + cb 6. Dot Product The dot product is defined by a = (a1 , a2 , a3 ), b = (b1 , b2 , b3 ) = a b = a1 b1 + a2 b2 + a3 b3 and obeys 0. 1. 3. 5. 7. a, b are vectors and a b is a number aa= a 0a=0 2 2. 6. ab =ba (ca) b = c(a b) ab = a b cos a (b + c) = a b + a c 4. a b = 0 a = 0 or b = 0 or a b In property 6, is the angle between a and b. 1 Cross Product The cross product is defined by a = (a1 , a2 , a3 ), b = (b1 , b2 , b3 ) = a b = (a2 b3 - a3 b2 , a3 b1 - a1 b3 , a1 b2 - a2 b1 ) and obeys 0. a, b and a b are all vectors in three dimensions 1. 2. a b a, b ab = a b sin 3. ^ = k, k = ^, k ^ = i ^ ^ ^ ^ i ^ i ^ 4. 5. 6. 7. 8. 9. ab = a b sin n ^ b a b = 0 a = 0 or b = 0 or a a b = -b a (ca) b = a (cb) = c(a b) a (b + c) = a b + a c a (b c) = (a b) c 10. a (b c) = (c a)b - (b a)c In properties 2 and 4, is the angle between a and b. In property 4, n = 1, n a, b and (a, b, n) obey the ^ ^ ^ right hand rule. 2 WARNING: Take particular care with properties 6 and 10. They are counterintuitive and cause huge numbers of errors. In particular, ab=ba a (b c) = (a b) c for most a, b and c. For example ^ i ^ (^ ) = ^ k = -k ^ = -^ i i ^ i ^ (^ ^) = 0 = 0 i i ^ ^ 3 ...
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This note was uploaded on 01/26/2011 for the course MATH 200 taught by Professor Unknown during the Spring '03 term at UBC.

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13.2-4 Properties of Vector Operations - Properties of...

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