prac1bsol

prac1bsol - MIT OpenCourseWare http:/ocw.mit.edu 18.02...

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

View Full Document Right Arrow Icon
MIT OpenCourseWare http://ocw.mit.edu 18.02 Multivariable Calculus Fall 2007 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .
Background image of page 1

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

View Full DocumentRight Arrow Icon
v v v v v v v v r a A p P 18.02 Practice Exam 1 B Solutions Problem 1. a) P = (1 , 0 , 0), Q = (0 , 2 , 0) and R = QP = ˆ ı QR = + 3 k ˆ . (0 , 0 , 3). Therefore −−→ and −−→ QP QR b) cos θ = v −−→ v · v −−→ v = a 1 , 2 , 0 A · a 0 , 2 , 3 A = 4 v QP vv QP v 1 2 + 2 2 2 2 + 3 2 65 v −−→ vv −−→ v Problem 2. a) −−→ = a− 1 , 2 , 0 A , −→ = a− 1 , 0 , 3 A . v v PQ PR v ˆ ı ˆ k ˆ v −−→ PQ × −→ = v 1 0 v = 6ˆ + 2 ˆ PR 2 ı + 3ˆ k . 1 0 3 Then area (Δ) = 2 1 v v PQ −−→ × PR −→ v v = 2 1 6 2 + 3 2 + 2 2 = 2 1 49 = 2 7 . b) A normal to the plane is given by −→ = PQ × PR a 6 , 3 , 2 A . N −−→ −→ = Hence the equation has the form 6 x + 3 y + 2 z = d . Since P
Background image of page 2
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 05/04/2011 for the course MATH 18.02 taught by Professor Auroux during the Spring '08 term at MIT.

Page1 / 2

prac1bsol - MIT OpenCourseWare http:/ocw.mit.edu 18.02...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online