s07wk02 - Math 23 B Dodson Week 2 Homework 12.2 vectors...

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Math 23B. DodsonWeek 2 Homework:12.2 vectors: unit, standard unit, notations12.3 dot product: orthogonal, proj, comp12.4 cross product: formula, propertiesProblem 12.2.19a:Find|a|anda-2bwhena=<6,2,3>, b=<-1,5,-2> .Solution:The length|a|=36 + 4 + 9 = 7,anda-2b=<6,2,3>-2<-1,5,-2>=<8,-8,7> .Problem 12.2.25:Find a unit vectoruthat has the same directionasa= 8i-j+ 4k.[variation/continuation: find a vector of length 4in the opposite direction.]Solution:The length|8i-j+ 4k|=64 + 1 + 16 =81 = 9,so the unit vector isu=1|a|a=198i-j+ 4k·=89i-19j+49k.
2Likewise, the vector with length 4 is-498i-j+ 4k·.Problem 12.3.23bc:Determine whether the given vectors are othogonal, parallel or neither.(b)a=<4,6>,b=<-3,2> .(c)a=-i+ 2j+ 5k,b= 3i+ 4j-k.Solution:(b)Recall that vectors areparallelif one isa scalar multiple of the other,a=cb,for a scalarc.This says<4,6>=c <-3,2>=<-3c,2c >,so 4 =-3c,and 6 = 2c.The second equation givesc= 3,but that’s not a solution of thefirst equation (withc=-43), so there’sno solutionc

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