Math 114 Quiz & HW No.3 Selected Solutions

Math 114 Quiz & HW No.3 Selected Solutions

Oct 2, 2009

Quiz 3, Sep 28:

1. Show that the equation represents a sphere, and find its center and radius:

2x2 + 2y2 + 2z2 = 8x - 24z + 1.

Completing squares in the equation gives 2(x2 - 4x + 4) + 2y2 + 2(x2 + 12x + 36) = 1 + 8 + 72 2(x - 2)2 + 2y2 + 2(z + 6)2 = 81 (x - 2)2 + y2 + (z + 6)2 = 81/2, which is an equation of a sphere with center (2, 0, -6) and radius 81/2 = 9/ 2.

2. a = i +2j -3k, b = -2i - j + 5k. Find |a - b|. |a - b|= | (i +2j -3k)- (-2i - j + 5k)| = |3i + 3j -8k | =

32 + 32 + (-8)2 = 82.

Quiz 3, Sep 30:

1. Show that the equation represents a sphere, and find its center and radius:

4x2 + 4y2 + 4z2 - 8x + 16y = 1.

Completing squares in the equation gives

4(x2 - 2x + 1) + 4(y2 + 4y + 4) + 4z2 = 1 + 4 + 16

4(x - 1)2 + 4(y + 2)2 + 4z2 = 21 (x - 1)2 + (y + 2)2 + z2 = 21/4, which

is an equation of a sphere with center (1, -2, 0) and radius

21 4

=

21 2

.

2. a = i +2j -3k, b = -2i - j + 5k. Find |a - b|. |a - b|= | (i +2j -3k)- (-2i - j + 5k)| = |3i + 3j -8k | =

32 + 32 + (-8)2 = 82.

HW3:

I graded 13.3 #20, 13.4 #36 for correctness and others for completion.

13.3 #20

|a| = 12 + 22 + (-2)2 = 9 = 3, |b| = 42 + 02 + (-3)2 = 25 = 5,

and a ? b = (1)(4) + (2)(0) + (-2)(-3) = 10. Thus

cos()

=

a?b |a||b|

=

10 3?5

=

2 3

,

=

cos-1(

2 3

)

13.4 #36

1

a = P Q =< -4, 2, 4 >, b = P R =< 2, 1, -2 > and c = P S =< -3, 4, 1 > .

a ? (b ? c) =

-4 2

-3

2 1 4

4 -2

1

= -4

1 4

-2 1

-2

2 -3

-2 1

+4

2 -3

1 4

=

-36 + 8 + 44 = 16, so the volume of the parallelepiped is 16 cubic units.

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