If your solution to stated problem does not match the given solution, you should self-critique per instructions at
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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.
010. Dot product, vector algebra
Question: `q001. The dot product of two vectors is equal to the product of the magnitudes of the vectors, multiplied by the cosine of the angle between the vectors. What is the dot product of vector v having magnitude 10 and angle 30 degrees, with vector w having magnitude 8 and angle 90 degrees?
Your solution:
Confidence Assessment:
Given Solution:
`aThe magnitudes are 10 and 8, and the angle between the vectors is the change in angle from 30 degrees to 90 degrees, or 60 degrees. The dot product is therefore
dot product = product of magnitudes * cos(angle) = 10 * 8 * cos(60 deg) = 40.
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Question: `q002. What are the x and y components of the vector v having magnitude 10 and angle 30 degrees, and vector w having magnitude 8 and angle 90 degrees?
What do you get if you add the product of the two x components to the product of the two y components?
How is this result related to the answer to the preceding exercise?
Your solution:
Confidence Assessment:
Given Solution:
`aThe vector v has x component vx = 10 cos(30 deg) = 8.7, approx., and vy = 10 sin(30 deg) = 5. The vector w has x component wx = 8 cos(90 deg) = 0 and y component wy = 8 sin(90 deg) = 8.
The product of the two x components is vx * wx = 8.7 * 0 = 0, and the product of the y components is vy * wy = 5 * 8 = 40.
The sum of these products is 0 + 40 = 40, which is identical to the result of the preceding exercise in which we found the dot product of v and w.
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Question: `q003. If vector v is represented by < v1, v2 > and vector w by < w1, w2 > then if the result of the preceding exercise is valid, how do we write in symbols the dot product of the two vectors?
In symbols how do we write the magnitudes of the two vectors?
How then do we write the statement that the dot product of the two vectors is equal to the product of the magnitudes of the vectors, multiplied by the cosine of the angle between the vectors?
Your solution:
Confidence Assessment:
Given Solution:
`aIf the preceding exercise generalizes then the dot product is the sum of the product of the x components and the product of the y components. In this case we would therefore say that the dot product is v1 * w1 + v2 * w2.
The magnitudes of the two vectors are | v | = sqrt(v1^2 + v2^2) and | w | = sqrt(w1^2 + w2^2).
The statement therefore says that
v1 * w1 + v2 * w2 = sqrt(v1^2 + v2^2) * sqrt(w1^2 + w2^2) * cos(theta).
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Question: `q004. Use the result of the preceding exercise to find the cosine of the angle between the vectors < 2, 3 > and < -7, 4 >.
What therefore is the angle between these vectors?
Your solution:
Confidence Assessment:
Given Solution:
`aThe cosine of the angle theta between the vectors is found using the fact that dot product is product of magnitudes multiplies by the cosine of the angle, expressed in detail as
v1 * w1 + v2 * w2 = sqrt(v1^2 + v2^2) * sqrt(w1^2 + w2^2) * cos(theta).
We easily rearrange the equation to get
cos(theta) = [ v1 * w1 + v2 * w2 ] / [ sqrt(v1^2 + v2^2) * sqrt(w1^2 + w2^2)].
In this case we have
cos(theta) = [ 2 * -7 + 3 * 4 ] / [ sqrt(2^3 + 3^2) * sqrt( (-7)^2 + 4^2) ] = -2 / ( sqrt(13) * sqrt(53) ] = -2 / 26.3 = .08 approx..
The angle is therefore the solution theta to the equation cos(theta) = .08, which gives us theta = 83 degrees, approx..
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Question: `q005. The vectors v = < 2, 4, 5 > and w = < -3, 7, 2 > exist in 3-dimensional space. What do you think are the magnitudes of these two vectors?
Your solution:
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Given Solution:
`aThe magnitude of a vector is found by taking the square root of the sum of the squares of its components, just as for a vector in 2-dimensional space. In this case we get magnitudes sqrt(2^2 + 4^2 + 5^2) = sqrt(45) = 6.7, approx., and sqrt((-3)^2 + 7^2 + 2^2) = sqrt(76) = 8.7 approx..
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Question: `q006. What is the dot product of the vectors v = < 2, 4, 5 > and w = < -3, 7, 2 > ? What therefore is the angle between these vectors?
Your solution:
Confidence Assessment:
Given Solution:
`aSince dot product = product of magnitudes * cos(theta) we have cos(theta) = dot product / product of magnitudes. In the preceding problem we found the magnitudes of the vectors to be sqrt(45) and sqrt(76). So if we can find the dot product we can find the cosine of the angle and therefore the angle.
The dot product is again the sum of the products of the invidual components, in this case 2 * (-3) + 4 * 7 + 5 * 2 = 32.
Thus we have
cos(theta) = 32 / ( sqrt(45) * sqrt(76) ) and
theta = arccos[ 32 / (sqrt(45) * sqrt(76) ) = 56.8 deg, approx..
Note that these vectors can actually be constructed in 3-dimensional space, and if the construction is accurate the angle will be as indicated.
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Question: `q007. Calculate the angle between the two vectors < 1, 7, -3, 4 > and < -3, -5, 2, 7 > .
Your solution:
Confidence Assessment:
Given Solution:
`aAs before we find that theta = arccos ( dot product / product of magnitudes ). In this case
theta = arccos [ (1 * -3 + 7 * -5 + -3 * 2 + 4 * 7) / ( sqrt(1^2 + 7^2 + (-3)^2 + 4^2 ) * sqrt( (-3)^2 + (-5)^2 + 2^2 + 7^2 ) ] =
arccos[ -16 / 80.8 ] =
101 degrees, approx..
Note that though these vectors occur in 4 dimensions and hence cannot be constructed in 3-dimensional space the calculation of the angle between them, extending in the most obvious manner the procedures of the preceding problems, give us a result with just a little additional calculation.