Math 200 - Exam 1 - 4/24/2013

NAME:

SECTION:

Directions:

• For the free response section, you must show all work. Answers without proper justifi-cation will not receive full credit. Partial credit will be awarded for significant progresstowards the correct answer. Cross off any work that you do not want graded.

• You have 50 minutes to complete this exam. When time is called, STOP WRITINGIMMEDIATELY.

• You may not use any electronic devices including (but not limited to) calculators, cellphones, or iPods.

Problem 1 Problem 2 Problem 3 Problem 4 Multiple Choice TOTAL14 Points 13 Points 13 Points 15 Points 45 Points 100 Points

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Free Response

For each of the following problems, you must show all of your work to earn fullcredit.

1. (14 Points) Consider the following planes:

P1 : x+ y + z = 7

P2 : 2x+ 4z = 6

Compute an equation of the line which passes through A(1, 2, 3) and is parallel to bothP1 and P2.

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2. (13 Points) Consider the line L : x = 1 + t, y = 3t, z = 1

(a) Find two points A and B on line L.

(b) Compute the distance from the point P (9, 4, 3) to the line L. (Hint: Consider

vectors−→AP and

−→AB.)

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3. (13 Points) The curves r1(t) = 〈t, t2, t3〉 and r2(t) = 〈sin t, sin 2t, t〉 intersect whent = 0. Find the angle between the tangent vectors to the curves at this point ofintersection. (You may leave your answer in terms of an inverse trigonometric function.)

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4. (3 Points Each) For each of the following, circle ”T” if the given statement is true or”F” if the given statement is false. If the statement is false, briefly explain why.

T F For any two vectors v and w in 3-space, v ×w = w × v.

T F For any vector v in 3-space, v × v is the zero vector.

T F If v and b are non-zero vectors, then (Projbv) · b = 0.

T F By definition, any lines L1 and L2 in 3-space which do not intersect are parallel.

T F The cross product of two unit vectors in 3-space is a unit vector.

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Multiple Choice

Circle the letter of the best answer. Make sure your circles inlude just one letter.These problems will be marked as correct or incorrect; partial credit will not beawarded for problems in this section. Each problem is worth 5 points.

5. Consider square ABCD with side length 4, shown below.

Which of the following is−→AB ·−→AC?

(a) 0

(b) 8

(c) 16

(d) 8√2

(e) 16√2

6. Suppose that u, v, and w are vectors in 3-space. Which of the following expressionsare meaningful?

I. (u+ v) ·w II. u+ (v ·w) III. (u · v) ·w IV. (u · v)w

(a) Only II

(b) Only III

(c) Only I and IV

(d) Only II and III

(e) I, II, III, and IV

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7. Let v = 〈2, 4〉 and w = 〈−2, 6〉. Which of the following is

∥∥∥∥1

2v −w

∥∥∥∥?

(a) 1

(b) 2

(c) 3

(d) 4

(e) 5

8. Which of the following is the equation of the plane which contains the point (3,−5, 2)

and is perpendicular to the line−→! (t) = 〈6,−3, 5〉+ t〈7,−4,−2〉?

(a) 7(x− 3)− 4(y + 5)− 2(z − 2) = 0

(b) 7(x+ 3)− 4(y + 5)− 2(z − 2) = 0

(c) 6(x− 3)− 3(y + 5) + 5(z − 2) = 0

(d) 6(x+ 3)− 3(y − 5) + 5(z + 2) = 0

(e) 3(x− 7)− 5(y + 4) + 2(z + 2) = 0

9. What is the domain of r(t) = ln (t+ 5)i− etj−√10− tk?

(a) t ≥ 10

(b) −5 < t ≤ 10

(c) −10 ≤ t < 5

(d) t ≤ −5 or t ≥ 10

(e) t ≤ −10 or t > 5

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10. Let v = 〈0, 7, 0〉 and let u be a vector of length 5 which starts at the origin and lies inthe xy-plane, as shown below.

What is the largest that ‖u× v‖ could possibly be?

(a) 1

(b) 12

(c) 30

(d) 35

(e) 140

11. Which of the following is a unit vector in the same direction as vector v = 〈√3, 3,−2〉?

(a) 4⟨√

3, 3,−2⟩

(b) 16⟨√

3, 3,−2⟩

(c)1

4

⟨√3, 3,−2

⟩

(d)1

16

⟨√3, 3,−2

⟩

(e)1

16

⟨−√3,−3, 2

⟩

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12. Which of the following is an equation of the sphere which has center at the point(1, 2, 3) and is tangent to the xz-plane?

(a) x2 + y2 + z2 = 1

(b) (x+ 1)2 + (y + 2)2 + (z + 3)2 = 1

(c) (x+ 1)2 + (y + 2)2 + (z + 3)2 = 2

(d) (x− 1)2 + (y − 2)2 + (z − 3)2 = 4

(e) (x− 1)2 + (y − 2)2 + (z − 3)2 = 14

13. Which of the following is a solution to the given initial value problem?

dr

dt= 〈t, 0, sin t〉

r(0) = 〈1, 2, 3〉

(a) r(t) = 〈1, 0, cos t〉

(b) r(t) = 〈1, 2, cos t+ 2〉

(c) r(t) =

⟨t2

2, 0,− cos t

⟩

(d) r(t) =

⟨t2

2+ 1, 2,− cos t+ 3

⟩

(e) r(t) =

⟨t2

2+ 1, 2,− cos t+ 4

⟩

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