#$&*
course Mth 277
9/6/2011 @ 11:13 p.m.
These questions range from easy to challenging. Don't expect to answer the all of the hardest questions correctly. Most students will not be able to do so. Everyone should be able to answer the easiest questions, but almost everyone will bog down at some point. Don't stay bogged down for too long before you move on to the next question. However do consider every question, and think about it for at least a couple of minutes.
Do explain how you get your results. Your instructor won't be able to tell much of anything from just a wrong answer, and even if the answer is right won't be able to tell from just the answer whether you got it by a process you can later build on.
You are also welcome to insert your own questions into the document. If you don't understand a problem, tell me as much as you can about what you think you do understand, and what you don't understand. The more information you give me the more likely I will be to be able to respond in a helpful way.
Insert each response on the line after **** and before #$&*. Submit using the Submit Work Form.
`q001. Consider the vector function
r(t) = 4 cos( pi / 12 * t) i + 4 sin(pi / 12 * t) j.
Sketch the vectors r (0), r(2), r(4) and r(6). Do the vectors 'move' in the clockwise or counterclockwise direction? Do they get longer, shorter or stay the same length?
****
The vectors move counterclockwise starting from the positive x-axis, and seem to stay mostly the same length, but possibly with slight growth.
#$&*
Sketch the path defined by r(t) for 0 <= t <= 6.
****
If I did it correctly, the path is a small arc from x=4 to about .11 above x=4.
#$&*
What are r( 3 ) and r( 4)?
****
R(3) = 3.99i + .055j and r(4) = 3.99i + .073j
#$&*
For the interval between t = 3 and t = 4, what is the average rate of change of r with respect to t?
****
The average rate of change of A with respect to B is (change in A)/(change in B). In this example r is the A and t is the B. So, the change in A is .018j, and the change in B is 1. .018j/1 = .018j/t (not knowing what the unit associated with t is)
#$&*
What is the expression for r ' (t)?
****
R(t) = -pi/3 * sin(pi/12 * t)i + pi/3 * cos(pi/12 * t)j + c
#$&*
What are the values of r ' (3), r ' (4) and r ' (3.5)?
****
R(3) = -.014i + 1.047j, r(3.5) = -.017i + 1.047j, r(4) = -.019i + 1.047j.
#$&*
How are your answers to the preceding question related to the average rate you calculated previously?
****
This question almost has the opposite effect as the previous, no rate of change found in the y-direction, and a small rate of change found in the x-direction.
#$&*
What is the expression for || r(t) || ?
****
Sqrt.((4cos(pi/12 * t))^2i + (4sin(pi/12 * t))^2)j
#$&*
What is the expression for r(t) / || (r(t) ||?
****
(4cos(pi/12* t)i + 4sin(pi/12 * t))j / Sqrt.((4cos(pi/12 * t))^2i + (4sin(pi/12 * t))^2j)
#$&*
What is the magnitude of the vector ( r(t) / || (r(t) || )?
****
This one wasnt immediately clear to me, but I plugged in 0 for t into the above equation and got the same for the top and bottom, so the magnitude appears to be 1.
#$&*
What is the dot product r(t) dot r ' (t) ?
****
[(-4pi/3( cos(pi/12 * t) * sin(pi/12 * t))i + (4pi/3( sin(pi/12* t) * cos(pi/12 * t))j]
#$&*
`q002. For the vector function
r(t) = 4 cos( pi / 12 * t) i + 8 sin(pi / 12 * t) j
Sketch the vectors r (0), r(2), r(4) and r(6). Do the vectors 'move' in the clockwise or counterclockwise direction? Do they get longer, shorter or stay the same length?
****
Counterclockwise, getting longer at twice the rate as the previous equation with a 4 in front of the sine function.
#$&*
Sketch the path defined by r(t) for 0 <= t <= 6.
****
A small arc from the x-axis at x=4 up to about .22 above the x-axis at x=4.
#$&*
What are r( 3 ) and r( 4)?
****
R(3) = 3.99i + .11j, r(4) = 3.99i + .15j.
#$&*
For the interval between t = 3 and t = 4, what is the average rate of change of r with respect to t?
****
The average rate of change of A with respect to B is (change in A)/(change in B). In this example r is the A and t is the B. So, the change in A is .04j, and the change in B is 1. .04j/1 = .04j/t (not knowing what the unit associated with t is)
#$&*
What is the expression for r ' (t)?
****
R(t) = -pi/3 * sin(pi/12 * t)i + 2pi/3 * cos(pi/12 * t)j + c
#$&*
What are the values of r ' (3), r ' (4) and r ' (3.5)?
****
R(3) = -.014i + 2.09j, r(3.5) = -.017i + 2.09j, r(4) = -.019i + 2.09j.
#$&*
****
#$&*
How are your answers to the preceding question related to the average rate you calculated previously?
****
This question almost has the opposite effect as the previous, no rate of change found in the y-direction, and a small rate of change found in the x-direction.
#$&*
What is the expression for || r(t) || ?
****
Sqrt.((4cos(pi/12 * t))^2i + (8sin(pi/12 * t))^2)j
#$&*
What is the expression for r(t) / || (r(t) ||?
****
(4cos(pi/12 * t)i + 8sin(pi/12 * t))j / Sqrt.((4cos(pi/12 * t))^2i + (8sin(pi/12 * t))^2j)
#$&*
What is the magnitude of the vector ( r(t) / || (r(t) || )?
****
Just like the above question, Im not extremely sure, but using the same logic, 1.
#$&*
What is the dot product r(t) dot r ' (t) ?
****
[(-4pi/3( cos(pi/12 * t) * sin(pi/12 * t))i + (16pi/3( sin(pi/12* t) * cos(pi/12 * t))j]
#$&*
`q003. For the vector function
r(t) = 4 cos( pi / 12 * t^2) i + 8 sin(pi / 12 * t^2) j
Find the values of t for which pi/12 * t^2 takes the values 0, pi/6, pi/3 and pi/2.
****
0, t = 0
Pi/6, t = 1.414 (or sqrt. 2)
Pi/3, t = 2
Pi/2, t = 2.449 (or sqrt. 6)
#$&*
Sketch the path defined by r(t) for 0 <= t <= sqrt(6).
****
A small arc from the x-axis at x=4 up to about .22 above the x-axis at x=4.
#$&*
What are r( sqrt(3) ) and r( 2)?
****
R(sqrt.(3)) = 3.99i + .11j, r(2) = 3.99i + .15j
#$&*
For the interval between t = sqrt(3) and t = 2, what is the average rate of change of r with respect to t?
****
The average rate of change of A with respect to B is (change in A)/(change in B). In this example r is the A and t is the B. So, the change in A is .04j, and the change in B is 1. .04j/1 = .04j/t (not knowing what the unit associated with t is)
#$&*
What is the expression for r ' (t)?
****
((-2pi/3 * t) * sin(pi/12 * t^2)) + ((4pi/3 * t) * cos(pi/12 * t^2)) + C
#$&*
What are the values of r ' (sqrt(3)), r ' (2) and r ' (sqrt(3.5))?
****
R(sqrt.(3)) = -.049i + 7.25j, r(2) = -.077i + 8.37j, r(sqrt.(3.5)) = -.063i + 7.84j
#$&*
How are your answers to the preceding question related to the average rate you calculated previously?
****
This time, the rate of both the x and y component of the vectors got larger, as opposed to the previous rate having just the j component change.
#$&*
What is the expression for || r(t) || ?
****
Sqrt.((4cos(pi/12 * t^2))^2i + (8sin(pi/12 * t^2))^2)j
#$&*
What is the expression for r(t) / || (r(t) ||?
****
(4cos(pi/12 * t^2)i + 8sin(pi/12 * t^2))j / Sqrt.((4cos(pi/12 * t^2))^2i + (8sin(pi/12 * t^2))^2j)
#$&*
What is the magnitude of the vector ( r(t) / || (r(t) || )?
****
Just like the above question, Im not extremely sure, but using the same logic, 1.
#$&*
What is the dot product r(t) dot r ' (t) ?
****
[(-4pi/3( cos(pi/12 * t^2) * sin(pi/12 * t^2))i + (16pi/3( sin(pi/12* t^2) * cos(pi/12 * t^2))j]
#$&*
`q004. For the foam piece you were given:
What are the lengths of the sides of the quadrilateral you traced on you paper, starting with the two sides that meet at your chosen point?
****
Going from the left side counterclockwise, and my point being between the first two lengths, 3.1 1xrcm (where 1xr = 1 times reduced), 5.6 1xrcm, 3.6 1xrcm, and 5.6 1xrcm.
#$&*
What is the altitude of the piece at your chosen point
****
6.8 1xrcm
#$&*
What are the altitudes of the piece at the other three corners of the quadrilateral you trace, listed in the counterclockwise order of the corners?
****
8.6 1xrcm, 5.4 1xrcm, and 3.5 1xrcm
#$&*
`q005. The sides of my quadrilateral, as drawn on the paper, were 4, 2.7, 4, 2.7. My quadrilateral didn't really form a perfect rectangle, but we'll assume it did.
The estimated altitudes of my four points, listed in the order specified above (going counterclockwise around the quadrilateral), were 5, 3, 2.5 and 4.6.
The vector forming the edge of my foam piece, above the side of the quadrilateral which has length 4 and intersects my initial point, has i component 4, with its k component running from altitude 5 to altitude 3. Thus the k component of this vector is 3 - 5 = -2. The vector is therefore 4 i - 2 k. What is the vector which forms the corresponding side of your quadrilateral?
****
5.6i - 3.3k
#$&*
The vector forming the edge of my foam piece, above the side of the quadrilateral which has length 2.7 and intersects my initial point, has j component 2.7, with its k component running from altitude 5 to altitude 4.6. Thus the k component of this vector is 4.6 - 5 = -.4. The vector is therefore 2.7 j - .4 k. What is the vector which forms the corresponding side of your quadrilateral?
****
3.1j + 1.8k
#$&*
Let's call these two vectors v and w. The cross product of my vectors is v X w = (4 i - 2 k) X (2.7 j - .4 k) = 5.4 i + 1.6 j + 10.8 k. The k component is the greatest, the j component the least, which is consistent with the direction of the vector perpendicular to the upper surface of the foam piece. What is the cross product of your two corresponding vectors?
****
10.23i - 10.08j + 17.37k
#$&*
That cross product is perpendicular to both of your two vectors v and w, so it should be perpendicular to the upper surface of the foam piece. Is the order of the magnitudes of your i, j and k components consistent with what you see when you point in the direction perpendicular to the surface?
****
Yes
#$&*
`q006. The paper has dimensions close to 28 cm x 21 cm. If you measured using a reduced ruler, each level of reduction increases these dimensions by a factor of about 1.6.
I estimate that a vector from the origin to the first point I marked on the paper is 9 i + 12 j. The point directly above this, when I placed the foam piece on the paper, was 5 cm above the paper, so the coordinates of that point were 9 i + 12 j + 5 k.
When I laid a flat piece of foam on my original foam piece, one of the points where its corner met the paper was at position 16 i + 24 j relative to the origin. The point was on the paper so its k coordinate was zero.
Estimate the coordinates of a point where a flat piece of material, laid across the top of your foam piece, would intersect the paper. If you left your foam piece behind, as many of you did, use your memory to estimate where such a point might lie.
****
15i + 9j in 1xrcm
#$&*
The vector from my point 9 i + 16 j + 5 k to the point 16 i + 24 j was (16 i + 24 j) - (9 i + 12 j + 5 k) = 7 i + 12 j - 5 k. What vector do you get when you do the same?
****
7.68i + 5.28j - 10.88k
#$&*
The flat piece used to obtain the second point was parallel to the top of the foam piece, so it should be perpendicular to the vector v X w. Two vectors are perpendicular if their dot product is zero. Calculating the dot product of these two vectors, we obtained a result of about 3. Both vectors had magnitudes greater than 10, so 3 is much less than the product of the two magnitudes, and is therefore quite close to 0.
****
Makes sense
#$&*
Do the same for your v X w vector and the vector between your two points on the paper. Is your result much smaller than the product of the magnitudes of these vectors?
****
Im getting large numbers when I multiply their magnitudes (22 times 14, so about 322, as a rough estimate) which doesnt seem right, though my numbers are quite a bit larger than yours.
#$&*
"
Self-critique (if necessary):
------------------------------------------------
Self-critique rating:
#$&*
course Mth 277
9/6/2011 @ 11:13 p.m.
These questions range from easy to challenging. Don't expect to answer the all of the hardest questions correctly. Most students will not be able to do so. Everyone should be able to answer the easiest questions, but almost everyone will bog down at some point. Don't stay bogged down for too long before you move on to the next question. However do consider every question, and think about it for at least a couple of minutes.
Do explain how you get your results. Your instructor won't be able to tell much of anything from just a wrong answer, and even if the answer is right won't be able to tell from just the answer whether you got it by a process you can later build on.
You are also welcome to insert your own questions into the document. If you don't understand a problem, tell me as much as you can about what you think you do understand, and what you don't understand. The more information you give me the more likely I will be to be able to respond in a helpful way.
Insert each response on the line after **** and before #$&*. Submit using the Submit Work Form.
`q001. Consider the vector function
r(t) = 4 cos( pi / 12 * t) i + 4 sin(pi / 12 * t) j.
Sketch the vectors r (0), r(2), r(4) and r(6). Do the vectors 'move' in the clockwise or counterclockwise direction? Do they get longer, shorter or stay the same length?
****
The vectors move counterclockwise starting from the positive x-axis, and seem to stay mostly the same length, but possibly with slight growth.
#$&*
Sketch the path defined by r(t) for 0 <= t <= 6.
****
If I did it correctly, the path is a small arc from x=4 to about .11 above x=4.
@& Between t = 0 and t = 6 the angle will change from 0 the pi / 2, so the cosine will change from 1 to 0, the sine from 0 to 1. So the vectors at t = 0 and t = 6 will be
`r(0) = 4 `i + 0 `j
`r(6) = 0 `i + 4 `j
The tip of the vector will move from (4, 0) to (0, 4).
You might have had your calculator in degree mode. You shouldn't use a calculator for angles which are multiples of pi/6 or pi/4.*@
#$&*
What are r( 3 ) and r( 4)?
****
R(3) = 3.99i + .055j and r(4) = 3.99i + .073j
@& For t = 3 and t = 4 the angle is pi/4 and pi/6, respectively.
You should know the sine and cosine of each of these angles.
You are almost certainly using a calculator in degree mode. Consider putting that thing away until you need it.*@
#$&*
For the interval between t = 3 and t = 4, what is the average rate of change of r with respect to t?
****
The average rate of change of A with respect to B is (change in A)/(change in B). In this example r is the A and t is the B. So, the change in A is .018j, and the change in B is 1. .018j/1 = .018j/t (not knowing what the unit associated with t is)
@& Good reasoning. You don't have the right vector components, but one you've corrected that you will get the right result.*@
#$&*
What is the expression for r ' (t)?
****
R(t) = -pi/3 * sin(pi/12 * t)i + pi/3 * cos(pi/12 * t)j + c
#$&*
What are the values of r ' (3), r ' (4) and r ' (3.5)?
****
R(3) = -.014i + 1.047j, r(3.5) = -.017i + 1.047j, r(4) = -.019i + 1.047j.
@& For t = 3.5 you will need that crutch (the calculator) so you can dust it off and use it here.
However be sure it's in degree mode.*@
#$&*
How are your answers to the preceding question related to the average rate you calculated previously?
****
This question almost has the opposite effect as the previous, no rate of change found in the y-direction, and a small rate of change found in the x-direction.
@& Think about this again when you get the right numbers. You have the right idea, but the right numbers will reinforce it better.*@
#$&*
What is the expression for || r(t) || ?
****
Sqrt.((4cos(pi/12 * t))^2i + (4sin(pi/12 * t))^2)j
@& It's Sqrt.((4cos(pi/12 * t)^2 + (4sin(pi/12 * t)^2)
SIgns of grouping have been modified. Also the `i and `j have been removed. They aren't part of the expression for the magnitude.
The 4's are also squared, which you probably intended.
Now, if expanded and simplified, the expression simplifies to just 4.
sin^2(theta) + cos^2(theta) = 1 for any theta.*@
#$&*
What is the expression for r(t) / || (r(t) ||?
@& As noted before, that denominator is just 4.*@
****
(4cos(pi/12* t)i + 4sin(pi/12 * t))j / Sqrt.((4cos(pi/12 * t))^2i + (4sin(pi/12 * t))^2j)
#$&*
What is the magnitude of the vector ( r(t) / || (r(t) || )?
@& Any time you divide a vector by its magnitude, the result has magnitude 1.*@
****
This one wasnt immediately clear to me, but I plugged in 0 for t into the above equation and got the same for the top and bottom, so the magnitude appears to be 1.
#$&*
What is the dot product r(t) dot r ' (t) ?
****
[(-4pi/3( cos(pi/12 * t) * sin(pi/12 * t))i + (4pi/3( sin(pi/12* t) * cos(pi/12 * t))j]
@& That need to be simplified.
You get 0.*@
#$&*
`q002. For the vector function
r(t) = 4 cos( pi / 12 * t) i + 8 sin(pi / 12 * t) j
Sketch the vectors r (0), r(2), r(4) and r(6). Do the vectors 'move' in the clockwise or counterclockwise direction? Do they get longer, shorter or stay the same length?
****
Counterclockwise, getting longer at twice the rate as the previous equation with a 4 in front of the sine function.
@& Not exactly at twice the rate, but while the preceding doesn't change in magnitude (i.e., doesn't get longer) this one does.*@
#$&*
Sketch the path defined by r(t) for 0 <= t <= 6.
****
A small arc from the x-axis at x=4 up to about .22 above the x-axis at x=4.
@& Again, put the calculator away for awhile. It's not needed to answer this question, and besides it's misleading you.*@
#$&*
What are r( 3 ) and r( 4)?
****
R(3) = 3.99i + .11j, r(4) = 3.99i + .15j.
#$&*
For the interval between t = 3 and t = 4, what is the average rate of change of r with respect to t?
****
The average rate of change of A with respect to B is (change in A)/(change in B). In this example r is the A and t is the B. So, the change in A is .04j, and the change in B is 1. .04j/1 = .04j/t (not knowing what the unit associated with t is)
#$&*
What is the expression for r ' (t)?
****
R(t) = -pi/3 * sin(pi/12 * t)i + 2pi/3 * cos(pi/12 * t)j + c
#$&*
What are the values of r ' (3), r ' (4) and r ' (3.5)?
****
R(3) = -.014i + 2.09j, r(3.5) = -.017i + 2.09j, r(4) = -.019i + 2.09j.
#$&*
****
#$&*
How are your answers to the preceding question related to the average rate you calculated previously?
****
This question almost has the opposite effect as the previous, no rate of change found in the y-direction, and a small rate of change found in the x-direction.
@& My notes on the last few problems parallel the notes I gave you on the first problem.*@
#$&*
What is the expression for || r(t) || ?
****
Sqrt.((4cos(pi/12 * t))^2i + (8sin(pi/12 * t))^2)j
#$&*
What is the expression for r(t) / || (r(t) ||?
****
(4cos(pi/12 * t)i + 8sin(pi/12 * t))j / Sqrt.((4cos(pi/12 * t))^2i + (8sin(pi/12 * t))^2j)
@& Get the grouping right, get rid of i and j, and then factor 4 out of the expression (by first factoring 16 out of the square root).*@
#$&*
What is the magnitude of the vector ( r(t) / || (r(t) || )?
****
Just like the above question, Im not extremely sure, but using the same logic, 1.
#$&*
What is the dot product r(t) dot r ' (t) ?
****
[(-4pi/3( cos(pi/12 * t) * sin(pi/12 * t))i + (16pi/3( sin(pi/12* t) * cos(pi/12 * t))j]
@& i and j are not part of the dot product, which is a pure number, not a vector.
This time it's not zero.*@
#$&*
`q003. For the vector function
r(t) = 4 cos( pi / 12 * t^2) i + 8 sin(pi / 12 * t^2) j
Find the values of t for which pi/12 * t^2 takes the values 0, pi/6, pi/3 and pi/2.
****
0, t = 0
Pi/6, t = 1.414 (or sqrt. 2)
Pi/3, t = 2
Pi/2, t = 2.449 (or sqrt. 6)
@& good*@
#$&*
Sketch the path defined by r(t) for 0 <= t <= sqrt(6).
****
A small arc from the x-axis at x=4 up to about .22 above the x-axis at x=4.
#$&*
What are r( sqrt(3) ) and r( 2)?
****
R(sqrt.(3)) = 3.99i + .11j, r(2) = 3.99i + .15j
#$&*
For the interval between t = sqrt(3) and t = 2, what is the average rate of change of r with respect to t?
****
The average rate of change of A with respect to B is (change in A)/(change in B). In this example r is the A and t is the B. So, the change in A is .04j, and the change in B is 1. .04j/1 = .04j/t (not knowing what the unit associated with t is)
@& You'll want to redo the above questions without the calculator. The angles you get are multiples of pi/6 and/or pi/4.*@
#$&*
What is the expression for r ' (t)?
****
((-2pi/3 * t) * sin(pi/12 * t^2)) + ((4pi/3 * t) * cos(pi/12 * t^2)) + C
#$&*
What are the values of r ' (sqrt(3)), r ' (2) and r ' (sqrt(3.5))?
****
R(sqrt.(3)) = -.049i + 7.25j, r(2) = -.077i + 8.37j, r(sqrt.(3.5)) = -.063i + 7.84j
@& Again you'll need the calculator for the last of the three.*@
#$&*
How are your answers to the preceding question related to the average rate you calculated previously?
****
This time, the rate of both the x and y component of the vectors got larger, as opposed to the previous rate having just the j component change.
#$&*
What is the expression for || r(t) || ?
****
Sqrt.((4cos(pi/12 * t^2))^2i + (8sin(pi/12 * t^2))^2)j
#$&*
What is the expression for r(t) / || (r(t) ||?
****
(4cos(pi/12 * t^2)i + 8sin(pi/12 * t^2))j / Sqrt.((4cos(pi/12 * t^2))^2i + (8sin(pi/12 * t^2))^2j)
#$&*
What is the magnitude of the vector ( r(t) / || (r(t) || )?
****
Just like the above question, Im not extremely sure, but using the same logic, 1.
#$&*
What is the dot product r(t) dot r ' (t) ?
****
[(-4pi/3( cos(pi/12 * t^2) * sin(pi/12 * t^2))i + (16pi/3( sin(pi/12* t^2) * cos(pi/12 * t^2))j]
#$&*
@& My notes would again parallel notes I gave earlier.*@
`q004. For the foam piece you were given:
What are the lengths of the sides of the quadrilateral you traced on you paper, starting with the two sides that meet at your chosen point?
****
Going from the left side counterclockwise, and my point being between the first two lengths, 3.1 1xrcm (where 1xr = 1 times reduced), 5.6 1xrcm, 3.6 1xrcm, and 5.6 1xrcm.
#$&*
What is the altitude of the piece at your chosen point
****
6.8 1xrcm
#$&*
What are the altitudes of the piece at the other three corners of the quadrilateral you trace, listed in the counterclockwise order of the corners?
****
8.6 1xrcm, 5.4 1xrcm, and 3.5 1xrcm
#$&*
`q005. The sides of my quadrilateral, as drawn on the paper, were 4, 2.7, 4, 2.7. My quadrilateral didn't really form a perfect rectangle, but we'll assume it did.
The estimated altitudes of my four points, listed in the order specified above (going counterclockwise around the quadrilateral), were 5, 3, 2.5 and 4.6.
The vector forming the edge of my foam piece, above the side of the quadrilateral which has length 4 and intersects my initial point, has i component 4, with its k component running from altitude 5 to altitude 3. Thus the k component of this vector is 3 - 5 = -2. The vector is therefore 4 i - 2 k. What is the vector which forms the corresponding side of your quadrilateral?
****
5.6i - 3.3k
#$&*
The vector forming the edge of my foam piece, above the side of the quadrilateral which has length 2.7 and intersects my initial point, has j component 2.7, with its k component running from altitude 5 to altitude 4.6. Thus the k component of this vector is 4.6 - 5 = -.4. The vector is therefore 2.7 j - .4 k. What is the vector which forms the corresponding side of your quadrilateral?
****
3.1j + 1.8k
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Let's call these two vectors v and w. The cross product of my vectors is v X w = (4 i - 2 k) X (2.7 j - .4 k) = 5.4 i + 1.6 j + 10.8 k. The k component is the greatest, the j component the least, which is consistent with the direction of the vector perpendicular to the upper surface of the foam piece. What is the cross product of your two corresponding vectors?
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10.23i - 10.08j + 17.37k
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That cross product is perpendicular to both of your two vectors v and w, so it should be perpendicular to the upper surface of the foam piece. Is the order of the magnitudes of your i, j and k components consistent with what you see when you point in the direction perpendicular to the surface?
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Yes
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`q006. The paper has dimensions close to 28 cm x 21 cm. If you measured using a reduced ruler, each level of reduction increases these dimensions by a factor of about 1.6.
I estimate that a vector from the origin to the first point I marked on the paper is 9 i + 12 j. The point directly above this, when I placed the foam piece on the paper, was 5 cm above the paper, so the coordinates of that point were 9 i + 12 j + 5 k.
When I laid a flat piece of foam on my original foam piece, one of the points where its corner met the paper was at position 16 i + 24 j relative to the origin. The point was on the paper so its k coordinate was zero.
Estimate the coordinates of a point where a flat piece of material, laid across the top of your foam piece, would intersect the paper. If you left your foam piece behind, as many of you did, use your memory to estimate where such a point might lie.
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15i + 9j in 1xrcm
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The vector from my point 9 i + 16 j + 5 k to the point 16 i + 24 j was (16 i + 24 j) - (9 i + 12 j + 5 k) = 7 i + 12 j - 5 k. What vector do you get when you do the same?
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7.68i + 5.28j - 10.88k
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The flat piece used to obtain the second point was parallel to the top of the foam piece, so it should be perpendicular to the vector v X w. Two vectors are perpendicular if their dot product is zero. Calculating the dot product of these two vectors, we obtained a result of about 3. Both vectors had magnitudes greater than 10, so 3 is much less than the product of the two magnitudes, and is therefore quite close to 0.
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Makes sense
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Do the same for your v X w vector and the vector between your two points on the paper. Is your result much smaller than the product of the magnitudes of these vectors?
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Im getting large numbers when I multiply their magnitudes (22 times 14, so about 322, as a rough estimate) which doesnt seem right, though my numbers are quite a bit larger than yours.
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@& Larger numbers will lead to much larger products. Seems reasonable to me.*@
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@& You're really doing well here, but you had a number of mostly-calculator-related avoidable errors. Check my notes.
Good work even considering the glitches.*@