Math 2043 Answer Key to the even-numbered review problems for Test 1 Spring 2015

Section 12.1

4. The projection of (2, 3, 5) onto the xy-plane is (2, 3, 0),onto the yz-plane is (0, 3, 5), onto the xz-plane is (2, 0, 5).

The length of the diagonal is√38.

12. Intersection with

xy-plane: a circle with center (2,−6, 0) and radius 3;xz-plane: no intersection;yz-plane: a circle with center (0,−6, 4) and radius

√21

Section 12.2

4. (a)−→AC (b)

−→CB (c)

−→DA (d)

−→DB 8. |w | =

√

| − u |2 + | − v |2 =√2

26. 〈−√6, 2

√6,√6〉

Section 12.3

64. (u+v) · (u−v) = 0 ⇒ u ·u−u ·v+v ·u−v ·v = |u |2 − |v |2 = 0 ⇒ |u |2 = |v |2 ⇒ |u | = |v |

Section 12.5

4. Vector equation: r = (14j− 10k)+ t(2i− 3j+9k), parametric equations: x = 2t, y = 14− 3t, z = −10+9t

26. 3x− y + 4z = 10

Chapter 12 Review

6.

⟨

7

3√6, 2

3√6, − 1

3√6

⟩

and⟨

− 7

3√6,− 2

3√6, 1

3√6

⟩

Section 13.2

40. (12te2t − 1

4e2t) i + (−t− ln |1− t|) j + arcsin tk + C 42. (1

2t2 + 1) i+et j+(tet − et + 2)k

Section 13.3

2.7

3; 6. 15

Section 16.2

34. mass = 1

2ka3

Section 14.3

34.∂w

∂x= yz2exyz,

∂w

∂y= xz2exyz,

∂w

∂z= (xyz + 1)exyz

Section 14.4

20. L(x, y) = 2 + 1 · (x− 1) + 1 · (y − 1), f(1.02, 0.97) ≈ L(1.02, 0.97) = 1.99

A12

2. 2 4. x = 3

2, y = − 1

2or x = 1

2, y = − 3

2

7. (a) 〈1,−5, 3〉 (b) both vectors are 0 (c)⟨

−22√566

, 1√566

, 9√566

⟩

or⟨

22√566

, −1√566

, −9√566

⟩

10. y = −1, z = −1 or y = 7

5, z = 1

5

A13

1. r(t) =(

2

5(t− 1)5/2 + 2

3(t− 1)3/2 − 1

15

)

i+(− tπ cos(πt) + 1

π2 sin(πt) +2

π ) j+(t lnx− t+ 5− 2 ln 2)k

2.

√5− 1

4