query 93

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course mth 277

9/14 11

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

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temporary disclaimer: Solutions to these problems were erroneously deleted and the problem solutions have been quickly reconstructed. These solutions are therefore not guaranteed, though the process by which they are obtained should be correct. So if you have discrepancies in arithmetic and other details, feel free to question the given solutions.

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_3

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Question: Find v dot w when v = 4i + j and w =3i + 2k.

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Your solution:

v*w=(4*3)+(1*3)=12+3=15

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Given Solution: v dot w = 4 * 3 + 1 * 2 = 12 + 2 = 14.

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Self-critique (if necessary):okay

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Self-critique rating:3

@& The given solution assumed vectors 4 `i + `j and 3 `i + `j, which is incorrect.

The correct answer, noticing the `k in the second vector, is just 12.*@

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Question: Determine whether v = 5i - 5j + 5k and w = 8i - 10j -2k are orthogonal.

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Your solution:

The two vectors are orthogonal if there is 90 degrees between them or when the dot product is equal to zero.

v*w= 5*8+-5*-10+5*-2=40+50-10=80

they are not orthogonal

confidence rating #$&*:

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Given Solution:

Two vectors are orthogonal if the angle between them is 90 deg, i.e., if and onlye if their dot product is zero.

The dot product of these vectors is 5 * 8 - 5 * (-8) + 5 * (-2) = 40 + 40 - 10 = 70.

They are not orthogonal.

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Self-critique (if necessary):you used -8 where it should be -10

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Self-critique rating:3

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Question: Find the angle between v = 2i +3 k and w = -j + 4k.

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Your solution:

Use the equation theta = arcCos( A dot B / || A || || B || )

v dot w =2*0+0*-1+3*4=12

|| v ||=sqrt(2^2+3^2)=3.6

|| w||=sqrt(-1^2+4^2)=4.12

theta = arcCos( 12 / (3.6*4.12) ) = arcos(.81)=35.9 degrees

confidence rating #$&*:

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Given Solution:

Since v dot w = || v || || w || cos(theta) we have

theta = cos^-1 ( v dot w ) || v || || w || = cos^-1 ( 10 / (sqrt(13) * sqrt( 17) ) = cos^-1 (.67) = 48 degrees, approx., or roughly.8 radians.

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Self-critique (if necessary):I think it was rounding issues

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Self-critique rating:3

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Question: Find two distinct unit vectors orthogonal to both v = i + 2j -2k and w = i + j - 2k.

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Your solution:

The dot product must equal 0

If you set both equal to zero you get and subtract equation 2 from equation1

(1) i+2j-2k=0

(2) i+j-2k=0

when you subtract them you get j=0

then plug j=0 into both equations you get

i+0-2k=0

i+0-2k=0

for both equations you get i=2k

both equations are equal to each other so they are perpendicular.

confidence rating #$&*:

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Given Solution:

Suppose a i + b j + c k is orthogonal to both. Then the dot product of this vector with each of the given vectors is zero, and we have

a + 2 b - 2 c = 0

a + b - 2 c = 0

Subtracting the second equation from the first we get b = 0.

With this value of b both our first and our second equation become

a - 2 c = 0

so that

a = 2 c.

Any vector of the form 2c i + c k is therefore orthogonal to our two vectors.

Any such vector has magnitude sqrt( (2 c)^2 + c^2) = sqrt( 5 c^2) = sqrt(5) | c |.

If c is positive then | c | = c and our vector is

(2 c i + c k ) / (sqrt(5) c) = 2 sqrt(5) / 5 i + sqrt(5) / 5 k.

If c is negative then | c | = - c and our vector will be

(2 c i + c k ) / (- sqrt(5) c) = - 2 sqrt(5) / 5 i - sqrt(5) / 5 k.

Our two solution vectors are equal and opposite. Each is a unit vector perpendicular to the two given vectors.

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Self-critique (if necessary):I didn’t go into great detail.

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Self-critique rating:3

@& You are stating that the vectors themselves are zero. This is not the case.

You haven't given a vector. You have the equation `i = 2 `k, which might imply a vector `i - 2 `k, but this vector is not orthogonal to either of he given vectors.*@

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Question: Let v = i - j + 4k and w = -i + 3j + 2k. Find cos(theta). Find s such that v is orthogonal to sv - w. Also find t such that v - tw is orthogonal to w

cos(theta) = v dot w / ( || v || || w ||) = 4 / (sqrt(18) sqrt(14) ) = 4 / (12 sqrt(7) ).

The condition v orthogonal to s v - w is

v dot (s v - w ) = 0

(i - j + 4 k ) dot ( (s - 1) i + (-s + 3) j + (4 s + 2) k ) = 0

which becomes

s - 1 + s - 3 + 16 s + 8 = 0

so that

18 s = 4

and

s = 4 / 18.

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Your solution: ??????????????? you gave us the solution as the question.

confidence rating #$&*:

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Given Solution:

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Self-critique (if necessary):okay

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Self-critique rating:3

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Question: Find the work performed when a force F = (6/11)i - (2/11)j + (6/11)k is applied to an object moving along the line from P(3,5,-4) to Q(-4,-9,-11).

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Your solution: pq= (-7i-14j-7k)

Fdotpq= (6/11)*-7+(-2/11)*-14+(6/11)*-7=-42/11+28/11-42/11=-56/11

confidence rating #$&*:

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Given Solution: The work is F dot `ds = ( (6/11)i - (2 / 11) j + (6 / 11) k ) dot (-7 i - 14 j - 7 k ) = -42/11 + 28 / 11 - 42 /11 = 28 / 11.

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Self-critique (if necessary):I think you answer is wrong I have checked my answer multiple times and im getting the same answer.

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Self-critique rating:3

@& You would still be expected to summarize the solution you got when you worked this problem out as part of the assigned problem set.*@

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