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February 5 Mathematics 206 Mr. Haines 2010 Multivariable Calculus Examination #
(15) I. If a = i + j and b = i - 2 j , compute these:
A. a • b =
B. || b || =
C. comp ba =
D. projba =
(5) II. Give an equation of the plane containing the points (1, 2, 3) and (3, 6, 7) and with normal 2 i - j.
(10) III. Give examples of the following sets in ጷ⡰
A. A set that is open and bounded.
B. A set that is open and not bounded.
(5) VI. Suppose a is a vector with tail at the point (1,2,3) and head at the point (3,5,5). Give a unit vector that is perpendicular to a.
(5) VII. Compute the area of the parallelogram in ጷ⡰^ with vertices (1,1), (5,7), (4,5), and (2,3).
(5) VIII. The plane P has coordinate equation 2ᡶ ㎗ 3ᡷ ㎗ ᡸ 㐄 5.
Give an equation for any line lying in P:
(10) IX. Give examples of:
A. Two unit vectors in ℜ^3 that are perpendicular.
B. Equations of two distinct parallel planes.
(10) XI. For the quadratic form
ᡨ䙦ᡶ, ᡷ, ᡸ䙧 㐄 ㎘ᡶ⡰^ ㎘ 2ᡷ⡰^ ㎘ 5ᡸ⡰^ ㎘ 2ᡶᡸ ,
A. Give a symmetric matrix S that is the matrix of this quadratic form.
B. By taking determinants and using Sylvester’s Theorem, determine if p is positive definite, negative definite, indefinite, or none of these.
(5) XII. A student says that any three points in ጷ⡱^ determine a plane. She wants to find the equation of the plane that contains the points (1, 1, 3), (1, 0, 4), and (1, -1, 5). She knows she needs to find a normal to the plane, but has trouble computing it. Why?
