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course Mth 277
June 20 round 9:45 pm
If your solution to stated problem does not match the given solution, you should self-critique per instructions athttp://vhcc2.vhcc.edu/dsmith/geninfo/labrynth_created_fall_05/levl1_22/levl2_81/file3_259.htm
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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.
qa 09_3
The dot product of two vectors is equal to the sum of the products of their components:
• If A = a_1 i + a_2 j + a_3 k andB = b_1 i + b_2 j + b_3 k, then A dot B = a_1 b_1 + a_2 b_2 + a_3 b_3.
The dot product is also equal to || A || * || B || cos(theta), where theta is the angle between the vectors. So
• A dot B = a_1 b_1 + a_2 b_2 + a_3 b_3
• A dot B = || A || * || B || cos(theta).
So for example if we know the components of A and B, we can easily find the dot product and the magnitudes of the two vectors. Having found the magnitudes and the dot product we can use the second relationship to get
cos(theta) = A dot B / || A || || B || so that
theta = arcCos( A dot B / || A || || B || ).
Two vectors are perpendicular to one another if the angle between them is 90 degrees. The cosine of 90 degrees is zero, and if the cosine of an angle between 0 and 180 degrees is zero the angle is 90 degrees. So two vectors are perpendicular if, and only if, their dot product is zero.
If the dot product of two vectors is zero we say that the vectors are orthogonal. In two or three dimensions, that means that the angle between the two vectors is 90 degrees.
Section 9.3
If you have a good Precalculus II background (or equivalent) with an appropriate introduction to vectors, or if you have worked through the recommended qa's on vectors, from Precalculus I, you should have little trouble with the query questions. So no q_a_ questions are included with this assignment.
For your convenience here's a listing of the Query questions:
1) 9.3.4 Find v dot u when v =<1,-5,0> u =<0,-4,2>.
Solution: v = < 1, -5, 0> u = < 0, -4, 2>
v * u = ( 1 * 0 + (-5) * (-4) + 0 * 2 )
= 0 + 20 + 0
= 20
2) 9.3.6 Find v dot w when v = 4i + j and w =3i + 2k.
Solution: v = 4j + j w = 3i + 2k
v * w = ( 4*3 + 1*0 + 0*2)
= 12 + 0 + 0
= 12
3) 9.3.10 Determine whether v = 5i - 5j + 5k and w = 8i - 10j -2k are orthogonal.
Solution: v = 5i - 5j + 5k w = 8i - 10j -2k
v * w = ( 5*8 + (-5)(-10) + 5(-2) )
= 40 + 50 - 10
= 80
Its not orthogonal because in order for it to be orthogonal v * w has to equal 0.
4) 9.3.12 Let v = 4i - 2j + k and w = -2i + j - k. Evaluate (v dot w) * w.
Solution: v = 4i - 2j + k w = -2i + j - k
= ( 4 * -2 + -2 * 1 + 1 * -1 ) * w
= ( -8 + -2 + -1) * w
= (-11) * w
= 22
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-11 * `w = -11 * (-2 `i + `j - `k) = 22 `i - 11 `j + 11 `k.
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5) 9.3.16 Find the angle between v = 2i +3 k and w = -j + 4k.
Solution: v * w = ( 2* 0 + 0 * -1 + 3 * 4)
= 0 + 0 + 12
= 12
Magnitude v = sqrt( 2^2 + 0^2 + 3^2)
= sqrt(13)
Magnitude w = sqrt( 0^2 + -1^2 + 4^2)
= sqrt( 0 + 1 + 16)
= sqrt(17)
Theta = cos ^-1 (v * w / magnitude (v) * magnitude (w) )
= cos ^-1 (12 / sqrt(13) * sqrt(17) )
Theta = 36.18
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36.18 would mean 36.18 radians. The default unit for angle is the radian.
What you clearly mean is 36.18 degrees; but when you mean degrees you have to specify degrees. 36.18 deg would be fine.
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6) 9.3.20 Find the scalar and vector projections of v = i - 2j onto w = j - 2k.
Solution: proj _w v = ( v* w / w * w) * w = 0 - 2 + 0 / 0 + 1 + 4 * (0 + 1 - 2)
= - 2/5 (0 + 1 - 2)
= -2/5j + 4/5k
Magnitude (v) * cos (theta) = v * w/ mag. (w) = (1 i - 2 j) - ( 1/ sqrt(5) j - 2/ sqrt(5) k )
= 1/ sqrt(5) + 4/ sqrt(5)
= 21 sqrt (5) / 5
7) 9.3.24 Find two distinct unit vectors orthogonal to both v = i + 2j -2k and w = i + j - 2k.
Solution: Unit vector = v / mag. (v) Unit Vector = w/ mag. (w)
= (1+2-2) / sqrt(9) = (1+1-2)/ sqrt(6)
= 1/3 = 0/ sqrt(6) = 0
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`v / || `v || is a vector, not a scalar.
`v = `i + 2 `j - 2 `k so || v || = 3, as you see.
So
`v / || `v || = ( `i + 2 `j - 2 `k ) / 3 = `i / 3 + 2/3 `j - 2/3 `k.
Similarly for `w / || `w ||.
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8) 9.3.30 Find x so that v = 2i - xj + 3k and w = -2i + j + xk are orthogonal.
Solution: if x = 2
v * w = (2 * -2 + -2 * 1 + 3 * 2)
= ( -4 + -2 + 6)
= 0
Its orthogonal if x = 2
9) 9.3.32 Give the direction cosines and direction angles of v = i - 4j.
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The direction cosines are the cosines of the angle made by the vector with the three coordinate axes.
The angles are denoted alpha, beta and gamma, the angles with the x, y and z axes respectively.
alpha is therefore the angle with the `i vector, and for vector v we have
cos(alpha) = `v dot `i / ( || `v || * || `i || )
In this case `v = `i - 4 `j, so `v dot `i = 1 and || `v || = sqrt(17). Of course || `i || = 1 so the direction cosine is
cos(alpha) = 1 / (sqrt(17) * 1) = sqrt(17) / 17.
See if you can now find the direction cosines for beta and gamma.
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Solution: I have no clue how to find the direction cosines and direction angles of a single vector; I ‘m confused on where to start.
10) 9.3.36(b,c,d) 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
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s is a scalar. So what is your expression for s `v - `w?
What does it mean for s `v - `w to be orthogonal to v?
This should give you an equation you can solve for s.
A similar strategy answers the second question.
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Solution:
11) 9.3.40 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).
Solution: W = F * D
D = P to Q
= < -4 - 3, -9 - 5, 11 - -4 >
= < -7, -14, 15 >
W = ( 6/11 - 2/11 + 6/11) * (-7 - 14 + 15)
W = -5.46
"
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Should be expressed as
W = ( 6/11 `i - 2/11 `j + 6/11 `k) dot (-7 `i - 14 `j + 15 `k)
Check your arithmetic on the dot product. I get 76 / 11, but I'm doing that in my head.
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Good overall, but you should submit revisions on a couple of these problems.
&#Please see my notes and submit a copy of this document with revisions, comments and/or questions, and mark your insertions with &&&& (please mark each insertion at the beginning and at the end).
Be sure to include the entire document, including my notes. &#
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