Prоblem 1. Let u→=-1,-2,4{"versiоn":"1.1","mаth":"u→=-1,-2,4"} аnd v→=-5,6,-7{"versiоn":"1.1","mаth":"v→=-5,6,-7"} be vectors. Part (a) (5 points) Find 2u→-v→{"version":"1.1","math":"2u→-v→"} Part (b) (5 points) Find u→{"version":"1.1","math":"u→"}, the magnitude of u→{"version":"1.1","math":"u→"} Part (c) (5 points) Find the unit vector in the direction of u→{"version":"1.1","math":"u→"} Problem 2. Let u→=2,2,-1{"version":"1.1","math":"u→=2,2,-1"} and v→=1,2,2{"version":"1.1","math":"v→=1,2,2"} be vectors. Part (a) (5 points) Find the dot product u→·v→{"version":"1.1","math":"u→·v→"} Part (b) (5 points) Find the cross product u→×v→{"version":"1.1","math":"u→×v→"} Part (c) (5 points) Find the angle between the two vectors u→{"version":"1.1","math":"u→"} and v→{"version":"1.1","math":"v→"} Problem 3. (15 points) Find an equation of the line L{"version":"1.1","math":"L"} passing through the points P(1,1,2){"version":"1.1","math":"P(1,1,2)"} and Q(1,-1,3){"version":"1.1","math":"Q(1,-1,3)"}, and then find the distance between the point R(0,0,5){"version":"1.1","math":"R(0,0,5)"} and the line L{"version":"1.1","math":"L"}. Problem 4. (15 points) Find an equation of the plane passing through the points P(1,-1,0){"version":"1.1","math":"P(1,-1,0)"}, Q(2,2,1){"version":"1.1","math":"Q(2,2,1)"}, and R(-1,-2,-1){"version":"1.1","math":"R(-1,-2,-1)"}. Problem 5. Consider the vector-value function r→(t)=t2,et,sint{"version":"1.1","math":"r→(t)=t2,et,sint"}. Part (a) (5 points) Find limt→0r→(t){"version":"1.1","math":"limt→0r→(t)"} Part (b) (5 points) Find the derivative of r→(t){"version":"1.1","math":"r→(t)"} Part (c) (5 points) Find ∫r→(t)dt{"version":"1.1","math":"∫r→(t)dt"} Problem 6. (10 points) Find the arch length of the curve r→(t)=2sint,2cost,t{"version":"1.1","math":" r→(t)=2sint,2cost,t"} with 1≤t≤3{"version":"1.1","math":"1≤t≤3"}. Problem 7. (15 points) Consider the curve r→(t)=t,t2,4{"version":"1.1","math":"r→(t)=t,t2,4"}. Find the unit tangent vector T→{"version":"1.1","math":"T→"} and the curvature at t=0{"version":"1.1","math":"t=0"}. Once you are done, please take pictures of your work, convert them into a pdf file; then scroll down to the bottom of this page to upload your file by clicking on "Add a File". Finally, please click "Submit Quiz." If you have trouble opening your file on your computer during the exam, you can also email your file to your instructor within 10 minutes after you submit your exam. Your instructor's email address is collier.gaiser@ccaurora.edu
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