Transcription of Midterm Exam I, Calculus III, Sample A
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Midterm Exam I, Calculus III, Sample A
Midterm Exam I, Calculus III, Sample A1. (10 points) Show that the 4 pointsP1= (0,0,0),P2= (2,3,0),P3= (1, 1,1),P4= (1,4, 1)are coplanar (they lie on the same plane), and find the equation of the plane that : u= P1P2= 2,3,0 ,v= P1P3= 1, 1,1 ,w= 1,4, 1 , and the scalar tripleproduct is equal tou (v w) = 2301 1114 1 = 2 114 1 3 111 1 + 0 1 114 = 0,so the volume of the parallelepiped determined byu,v, andwis equal to 0. This means thatthese vectors are on the same plane. So,P1,P2,P3, andP4are (10 points) Find the equation of the plane that is equidistant from the pointsA= (3,2,1)andB= ( 3, 2, 1) (that is, every point on the plane has the same distance from the twogiven points).Solution:The midpoint of the pointsAandBis the pointC=12(3,2,1)+( 3, 2, 1)]=(0,0,0). A normal vector to the plane is given by AB= 3,2,1 3, 2, 1 = 6,4,2.
So, the equation of the sphere is (x 22) 2+y +(z ( 2)) = 42, the center is (3;0; 2) and radius 4. 2.(6 Points) Write each combination of vectors as a single vector
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