EXAM 1

Math 212, 2018 Summer Term 1, Clark Bray.

YOU MUST SHOW ALL WORK AND EXPLAIN ALL REASONING TO RECEIVE CREDIT.CLARITY WILL BE CONSIDERED IN GRADING.

No notes, no books, no calculators.

All answers must be simplified.

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1. (20 pts) In this question we consider the list of vectors �v = (2, 1, 3), �w = (0, 1, 1), �x = (3, 5, 1).

(a) Compute the area of the parallelogram defined by �v and �w.

(b) Use your work from part (a) to find the volume of the parallelepiped defined by �x, �v, �w.

(c) Use your work from part (b) to decide if the list �v, �w, �x is in right hand order.

(extra space for question from other side)

2. (20 pts)

(a) A plane P contains the points �a, �b, and �c (not all on any single line). Suppose �n is computed

as �n = (�b−�a)× (�c−�a). Show directly (using algebraic properties of dot and cross products)that the outputs from the function

�x(s, t) = �a+ s(�b− �a) + t(�c− �a)

all satisfy the equation �n · �x = �n · �a.

(b) Use the result of part (a) to find a parametrization of the plane that goes through the points(1, 0, 1), (4, 3, 2), and (4, 5, 6).

(extra space for question from other side)

3. (20 pts)

(a) The surface S is obtained by applying to the unit sphere the following sequence of modifica-tions, in the listed order: (1) translate in the positive x-direction by a distance 2; (2) stretchsimultaneously in all directions from the origin by a factor of 3; (3) reflect over the yz plane.

Find the equation of S.

(b) Draw the surface with equation�x2 + y2 = 1− z4, and identify any vertices it might have.

(extra space for question from other side)

4. (20 pts) Planet X-212 rotates around its sun (located at the origin) in the xy-plane in a circularorbit in the positive θ direction, moving at a constant speed. Using certain units, the radius ofits orbit is 50 and the period (amount of time it takes to go around the sun once) is 20. Its moon,X-212-M, orbits X-212 (also in a circular orbit in the positive θ direction, at a constant speed)with radius 3 and period 4. At time t = 0, X-212 is at (0, 50) and X-212-M is at (3, 50).

(a) Find the parametrization of X-212’s path that gives position in terms of time.

(b) Find the parametrization of X-212-M’s path that gives position in terms of time.

(extra space for question from other side)

5. (20 pts) The 3×3 matrices R, S, and Q = RS have rows, columns, and elements as noted below.

R =

− �R1 −− �R2 −− �R3 −

=

| | |�r1 �r2 �r3| | |

=

312 141 253534 623 137234 478 812

S =

− �S1 −− �S2 −− �S3 −

=

| | |�s1 �s2 �s3| | |

=

652 432 821144 287 458294 149 151

Q =

− �Q1 −− �Q2 −− �Q3 −

=

| | |�q1 �q2 �q3| | |

(a) Write �Q2 as a linear combination of the rows of either R or S.

(b) Write �q3 as a linear combination of the columns of either R or S.

(extra space for question from other side)