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course Mth 277
June 20 11:16 pm
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.
qa 09_04
Section 9.4
The cross product of two vectors A and B is a vector perpendicular to the two vectors, the direction determined by the right-hand rule, and its magnitude is || A || * || B || * sin(theta), where theta is the angle between the two.
The formula for the cross product is a little more complicated than the simple formula for the dot product, but it's not bad:
A `X B is the determinant of the matrix whose first row is [ i, j, k ], whose second row is [ A_1, A_2, A_3 ], the coefficients of the A vector, and whose third row is [ B_1, B_2, B_3 ], the coefficients of the B vector.
Experience shows that students who have succeeded on Assignments 1-3 typically do not require a q_a_ for this assignment, and may move directly to the Query.
For your convenience here's a listing of the Query questions:
1) 9.4.6 Find v X w when v = 10i - 2j + 4k and w =-i -(1/2)j - 3k. (Where X denotes the cross product)
Solution: v = 10i -2j + 4k w = -1 - (1/2)j -3k
v X w = ( u_2*v_3 - u_3*v_2)I - (u_1*v_3 - u_3*v_1)j + (u_1*v_2 - u_2 * v_1)k
= 8i + 26j -7k
2) 9.4.10 Find v X w when v = sin(theta)i + cos(theta)j and w = -cos(theta)i + sin(theta)j (theta is any angle).
Solution: v = sin(theta)i + cos(theta)j and w = -cos(theta)i + sin(theta)j
v X w = ( u_2*v_3 - u_3*v_2)I - (u_1*v_3 - u_3*v_1)j + (u_1*v_2 - u_2 * v_1)k
v X w = 1 k
3) 9.4.12 Find sin(theta) where theta is the angle between v = -i + j and w = -i + j + 2k.
Solution: mag. (v) = sqrt( (-1) ^2 + (1) ^2 )
= sqrt (2)
Mag (w) = sqrt( (-1) ^2 + (1) ^2 + (2) ^2 )
= sqrt (6)
Theta = cos ^-1 (1 + 1 + 0) / sqrt (2) * sqrt (6)
Theta = cos ^ -1 ( 2/ sqrt(2) * sqrt (6) = 54.74
Sin( theta) = Sin (54.74) = .817
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You could also have used the cross product, which directly involves the sine.
Note also that sin(cos^-1(x)) = sqrt(1 - x^2), either by the Pythagorean identity or by the fact that cos^-1(x) is the angle whose cosine is x, which would be the angle of a triangle with hypotenuse 1 and adjacent side x. The opposite side, by the Pythagorean Theorem, would be sqrt( 1 - x^2 ), giving us the expression for the sine.
Your result .817 is a decent approximation to a result that can be expressed exactly, but is still an approximation.
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4) 9.4.16 Find a unit vector which is orthogonal to both v = -i + 3j and w = i - j - k
Solution:
5) 9.4.18 Find a unit vector which is orthogonal to both v = 2i - j and w = 2j - k.
Solution:
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How do you find a vector orthogonal to two given vectors?
How do you find a unit vector in the direction of a specified vector?
How therefore do you find a unit vector orthogonal to these two vectors?
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6) 9.4.22 Find the area of the parallelogram determined by the vectors v = 4i + k and w = 4j - k.
Solution: v = 4i + k and w = 4j - k.
Mag. (u X v) = mag. (u) * mag. (v) * mag. (sin (theta)) * mag. (n) = mag. (u) * mag. (v) sin (theta)
Mag. (u X v) is the area of a parallelogram
Mag. (u X v) = sqrt. ((-4) ^2 + (4) ^2 + (16) ^2)
= sqrt. (256)
= 16
7) 9.4.24 Find the area of the triangle with vertices P(2,0,0), Q(1,1,-1), R(3,1,2).
Solution: P to Q = (1-2, 1-0, -1-0) = (-1, 1, -1)
P to R = (3-2, 1-0, 2-0) = (1, 1, 2)
Mag. ( P to Q * P to R) = Mag. ( 3i + j + 2k)
= sqrt ( (3) ^2 + (1) ^2 + (2) ^2 )
= sqrt (14)
The area of a triangle PQR is half of absolute ( P to Q x P to R) = sqrt (14) / 2
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The area of the triangle is || `PQ X `PR ||.
For example || `PQ || sin(theta) could be regarded as the altitude with || `PR || being the base.
Alternatively we could use
|| `PQ X `QR ||
or
|| `PR X `QR ||.
In any of these expressions, either vector can be regarded as the base and the magnitude of the other vector, multiplied by sin(theta), as the altitude.
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8) 9.4.28 Determine if each of the following products is a vector, scalar, or not defined at all. Explain why. u X (v X w) , u dot (v dot w), (u X v) dot (w X r).
Solution: u X (v X w) is a vector
u X (v X w) = (u dot w) v - (u dot v) w
u dot (v dot w) is a scalar
(u X v) dot (w X r) is not defined at all…you can’t multiply two vectors by two scalars
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u X v is a vector
w X r is a vector
So you can calculate the dot product of the two vectors, which gives you a scalar.
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9) 9.4.30 Find the area of the parallelepiped determined by u = i - j, v = i - 2k, and w = 4k
Solution:
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Consider the base to be defined by `v X `w. The area of the base is thus || `v X `w ||.
The component of `u perpendicular to the base will therefore be the altitude.
`v X `w is a vector perpendicular to the base. What is the component of `u parallel to this vector?
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10) 9.4.38 Find a number t such that the vectors -i - j, i - (1/2) j + (1/2)k and -2i -2j - 2tk all lie in the same plane.
Solution:
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Three vectors lie in the same plane provided one is perpendicular to the cross product of the other two.
As you know two vectors are perpendicular if their dot product is zero.
There are other possible conditions as well.
The condition that the vectors lie in the same plane needs to be written down and translated into an equation or a system of equations.
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11) 9.4.39 u = 2i + 2j, v= i-(1/2)j+ (1/2)k, w = i. Compute (u X v) X w and u X (v X w). What does this say about the associativity of the cross product?
Solution: u = 2i + 2j, v= i-(1/2)j+ (1/2)k, w = i.
(u X v) = i - j - 3k
X w = -3j + k
(v X w) = 1/2j + 1/2k
u X (1/2j + 1/2k) = -i + j - k
The associatively of the cross product shows that both solutions are not the same. In the first solution there is no i vector and in the second solution all vectors are present.
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And thus the cross product is not associative.
Good.
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You do need to revise and/or extend a couple of your solutions.
&#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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