{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Math206Test1 - Mid-term Exam 1 MthSc 206 013 Calculus of...

Info icon This preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
Mid-term Exam # 1 September 24, 2007 MthSc 206 – 013: Calculus of Several Variables Instruction: PRINT your name clearly in capitals and underline your last name . NAME: EXAM SOLUTION 1. (3pts). Show that the following equation represents a sphere and find its center and radius: x 2 + y 2 + z 2 + 2 x - 6 y + 4 z + 10 = 0 . Solution: x 2 + 2 x + y 2 - 6 y + z 2 + 4 z = - 10 x 2 + 2 x + 1 + y 2 - 6 y + 9 + z 2 + 4 z + 4 = - 10 + 1 + 9 + 4 ( x + 1) 2 + ( y - 3) 2 + ( z + 2) 2 = 4 It is an equation for a sphere. The center is c ( - 1 , 3 , - 2) and its radius is 2 . 2. (8pts) Two vectors a = 1 , 2 , 3 and b = 0 , 1 , 0 are given. (a) (3pts) Find the scalar projection (scalar component) of b onto a . Solution: Comp a b = a b | a | = 1 , 2 , 3 0 , 1 , 0 q (1) 2 + ( 2) 2 + ( 3) 2 = 2 6 = 3 3 = 0 . 577 . (b) (3pts) Find the vector projection of b onto a . Solution: Proj a b = Comp a b · a | a | = 3 3 · 1 , 2 , 3 6 = 1 3 2 1 , 2 , 3 (c) (2pts) Find the angle between the vectors a and b . Solution: Since a b = | a | | b | cos θ , we have 1 , 2 , 3 0 , 1 , 0 = 6 · 1 · cos θ 2 = 6 · 1 · cos θ.
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern