query_092

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course MTH 277

7/13/2012 320AM

If your solution to stated problem does not match the given solution, you should self-critique per instructions at

http://vhcc2.vhcc.edu/dsmith/geninfo/labrynth_created_fall_05/levl1_22/levl2_81/file3_259.htm

.

Your solution, attempt at solution. If you are unable to attempt a solution, give a phrase-by-phrase interpretation of the problem along with a statement of what you do or do not understand about it. This response should be given, based on the work you did in completing the assignment, before you look at the given solution.

At the end of this document, after the qa problems (which provide you with questions and solutions), there is a series of Questions, Problems and Exercises.

query_09_2

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Question: Find u + v, u - v, (5/2)u, and 2u + 3v for the following vectors: u = <1,2,-3>, v = < -1,-2,3>.

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Your solution:

u + v = <0,0,0>

u - v = <2,4,-6>

(5/2)u = < 5/2, 5, -15/2>

2u + 3v = <2, 4, -3> + <-3, -6, 9> = <-1, -2, 6>

confidence rating #$&*:

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Given Solution:

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Self-critique (if necessary): OK

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Self-critique rating:

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Question: Find the standard form equation of the sphere with center (-1,2,4) and radius 2.

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Your solution:

(x-h)^2 + (y-k)^2 + (z-n)^2 = r^2

(x+1)^2 + (y-2)^2 + (z-4)^2 = 4

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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Given Solution:

A point (x, y, z) is on the given sphere if its distance from (-1, 2, 4) is 2, so that

sqrt( (x - (-1))^2 + (y - 2)^2 + (z - 4)^2 ) = 2

and

(x + 1)^2 + (y - 2)^2 + (z - 4)^2 = 4.

This is the equation of the sphere in one form.

Expanding the squares we obtain

x^2 + 2 x + 1 + y^2 - 4 y + 4 + z^2 - 8 x + 16 = 4

which we rearrange to the standard form

x^2 + 2 x + y^2 - 4 y + z^2 - 8 z + 13 = 0.

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Self-critique (if necessary):

I didn't put the equation in standard form. I got halfway, but didn't expand. I now understand that standard form is when I expand the equation out.

I don't understand how you got 13?????????????????? I worked out the solution, but I got 17.

(x + 1)^2 + (y - 2)^2 + (z - 4)^2 = 4.

x^2 + 2 x + 1 + y^2 - 4 y + 4 + z^2 - 8 x + 16 = 4

x^2 + 2 x + y^2 - 4 y + z^2 - 8 x + 21 = 4

x^2 + 2 x + y^2 - 4 y + z^2 - 8 z + 17 = 0.

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Self-critique rating:3

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You are again correct. 21 - 4 = 17.

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Question: Find the center and radius of the sphere with equation x^2 + y^2 + z^2 - 2x - 6y + 12z - 17 = 0.

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Your solution:

x^2 + y^2 + z^2 - 2x - 6y + 12z - 17 = 0

x^2 -2x +1 + y^2 -6y +9 + z^2 + 12z +36 = 17 +1 + 9 + 36

(x-1)^2 + (y-3)^2 +(z+6)^2 = 63

radius is 3sqrt(7) or approximately 7.94 units

center of the sphere is at (1,3,-6)

confidence rating #$&*:

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Given Solution:

Completing the squares we obtain

(x^2 - 2 x + 1 - 1) + (y^2 - 6 y + 9 - 9) + (z^2 + 12 z + 36 - 36) = 17

which can be written as

(x - 1)^2 - 1 + (y - 3)^2 - 9 + (z + 6)^2 - 36 = 17

and finally as

(x - 1)^2 + (y - 3)^2 + (z + 6)^2 = 63

This sphere is centered at (1, 3, -6) and has radius sqrt(63) = 3 sqrt(7).

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Self-critique (if necessary):

BAM! oh yeh!

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Self-critique rating:3

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Question: Find the standard representation and length of PQ when P = (-3,1,4) and Q = (2,-4,-3).

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Your solution:

PQ= <5, -5, -7>

ll PQ ll = sqrt(25+25+49)= sqrt(99) = 3(sqrt11)

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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Given Solution:

PQ = (2 - (-3) ) i + (-4 - 1) j + (-3 - 4) k = 5 i - 5 j - 7 k.

|| PQ || = sqrt( 5^2 + 5^2 + 7^2) = sqrt(99).

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Self-critique (if necessary): OK

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Self-critique rating:

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Question: Find a unit vector in the direction of v = <-1, sqrt(3), 4>.

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Your solution:

ll v ll = sqrt(1+3+16) = sqrt(20) = 2sqrt(5)

find unit vector [1/2sqrt(5)] * v = < -1/2sqrt(5), sqrt(3)/2sqrt(5), 2/sqrt(5)>

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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Given Solution:

|| v || = sqrt( 1^2 + sqrt(3) ^ 2 + 4^2 ) = sqrt( 26 )

so a unit vector in the direction of v is

v / || v ||= < -1, sqrt(3), 4 > / sqrt(26) = <-sqrt(26) / 26, sqrt(78) / 26, 4 sqrt(26) / 26)> .

4 sqrt(26) / 26 is 2 sqrt(26) / 13.

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Self-critique (if necessary):

the magnitude is sqrt(20). NOT sqrt(26). 1+3+16= 20 NOT 26.

How do I simplify my answer??????????? I don't know how to simplify sqrt(3)/2sqrt(5)

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Self-critique rating: 3

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sqrt(3) / (2 sqrt(5) )

= sqrt(5) / sqrt(5) * sqrt(3) / (2 sqrt(5))

= sqrt(3) sqrt(5) / (2 sqrt(5) sqrt(5))

= sqrt(3) sqrt(5) / 10

When you have a square root in the denominator, multiply both numerator and denominator by that square root to put in standard form. Standard form doesn't have square roots in the denominator.

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