sectionNineTwo

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course MTH 277

09/14/20119:36 am

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, -6> + <-3, -6, 9> = <-1, -2, 3>

confidence rating #$&*:232; 3

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Given Solution:

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Self-critique (if necessary):

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

confidence rating #$&*:232; 3

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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):

Didn’t pay enough attention to the question. Got the initial equation right, but didn’t shift it to standard form

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Self-critique rating: 3

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

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

(x - 1)^2 + (y-3)^2 + (z + 6)^2 = 62

center = (1, 3, -6)

radius = sqrt(62)

confidence rating #$&*:232; 2

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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):

Minor arithmetic error when calculating the total from the completed squares

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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:

Length = sqrt(25 + 25 + 49)

Sqrt(99)

Length = sqrt(99)

Standard representation: (5, -5, 7)

confidence rating #$&*:232; 3

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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):

Forgot to put the i, j, and k after the appropriate numbers

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Self-critique rating: 3

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

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Your solution:

That is the unit vector

confidence rating #$&*:232; 1

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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):

I completely brain-farted on this one. It mistook one term for another.

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Self-critique rating:2

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Question: Sketch and describe the cylindrical surface given by y = cos x.

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Your solution:

An endless wave. Almost like a sound diagram.

confidence rating #$&*:232; 2

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Given Solution:

In the x-y plane y = cos(x) consists of a sinusoidal function oscillating between the lines y = -1 and y = 1, with period 2 pi radians, and containing the point (0, 1).

The surface in 3 dimensions repeats this same curve for every value of z, so that the graph represents a wavy curtain hanging vertically downward, intersecting the xy plane along the sinusoidal curve.

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Self-critique (if necessary):

I gave the basic shape, but gave no details, nor direction. Needed more explanation.

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Self-critique rating: 2

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Question: Determine if u = 2i + 3j + -4k is parallel to v = <1,-3/2,2>.

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Your solution:

no

@& Of course you need to justify a 'yes' or 'no' answer.*@

confidence rating #$&*:232; 3

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Given Solution:

Two vectors are parallel if the angle between them is 0 or pi radians (180 degrees), meaning that the cosine of the angle between them is 1 or -1.

u dot v = || u || || v || cos(theta)

so that

cos(theta) = u dot v / (|| u || || v || )

= (2 * 1 + 3 * (-3/2) + (-4 * 2) ) / ( sqrt(2^2 + 3^2 + 4^2) * sqrt( 1^2 + (3/2)^2 + 2^2) )

= (-21/2) / (sqrt( 29) sqrt(29/4).

This is not 1 or -1, so the cosine is neither 0 nor pi rad (i.e., 180 deg).

The vectors are therefore not parallel.

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Self-critique (if necessary): OK

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Self-critique rating:OK

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Question: Find the lengths of the sides of the triangle and determine if the triangle with vertices A(3,0,0), B(7,1,4) and C(5,4,4) is a right triangle, isosceles triangle, both, or neither.

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Your solution:

AB = sqrt(33)

BC = sqrt(13)

CA = 6

neither

confidence rating #$&*:232; 3

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Given Solution:

The sides can be represented by the vectors

AB = < 4, 1, 4 >,

BC = < -2, 3, 0 > and

AC = < 2, 4, 4 >.

The magnitudes of these vectors are respectively

sqrt(33)

sqrt(13)

sqrt(36).

None of the sides are the same length so the triangle is not isosceles.

The sum of the squares of the shorter two side is 33 + 13 = 46, which is not equal to the sum of the longest, so the triangle is not a right triangle.

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Self-critique (if necessary): OK

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Self-critique rating:OK

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#*&!

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