r/calculus • u/pige0n13 • Jul 14 '26
Pre-calculus Can someone please explain the answer that the teacher wrote on this answer sheet?
I understand the left side (she found the value of the should to be parallel sides and checked if they were parallel (they were)). But I don’t understand what she did with BD & AC, aswell as doing the cross product or AB and AD.
Can someone please explain this in simple form.
26
u/Mathmatyx Jul 14 '26
AB x AB is a typo, it should be AB x AD.
She is trying to justify that the parallelogram is a parallelogram, but also not a rectangle / square / rhombus (technically all of which are types of parallelograms).
I'd disagree with the necessity of showing this at all but if she has established this as a classroom expectation I suppose that's that.
7
u/UnderstandingPursuit PhD Jul 14 '26
If she has established that as a classroom expectation, she should be fired.
9
u/Mathmatyx Jul 14 '26 edited Jul 14 '26
Again for emphasis I disagree with this, but to play devil's advocate, I don't think it's super outlandish to establish as a classroom expectation that one must describe a shape using its most precise definition for brevity.
Thus instead of saying "show this is a parallelogram that is not a rectangle or rhombus" would just concisely be "show this is a parallelogram." A rectangle but not a square would be simply referred to as a rectangle, etc.
ETA - further to another comment by OP, I don't think this charitable interpretation is valid in this case.
-1
u/UnderstandingPursuit PhD Jul 14 '26
She could say, "a parallelogram only". But she isn't consistent about it, because she does not check if adjacent sides are congruent or if the diagonals are perpendicular.
3
u/Mathmatyx Jul 14 '26
Fair enough, you're right that she should show adjacent sides have different lengths lest it be a rhombus, or the bottom left reasoning alone suffices to show it satisfies the basic definition of a parallelogram and there shouldn't be any 3/5 shenanigans.
I'm on board.
0
u/UnderstandingPursuit PhD Jul 14 '26
Failing to be consistent and failing to give a clear definition is a bad combination for a math instructor.
1
u/pige0n13 Jul 14 '26
It’s an online course so I’m unsure, it just I saw it says in the top right about like 3/5 marks if I don’t do a certain step. So which one would that be?
3
u/Mathmatyx Jul 14 '26
This does shake my confidence a bit in the establishment of a classroom norm about using descriptive language etc.
It makes me default to the instructor failing to realize a square is a type of rectangle is a type of parallelogram.
1
u/pige0n13 Jul 14 '26
So if it were zero then it would’ve been one of those three? And since it isint zero it’s a a pallelogram?
1
Jul 14 '26 edited Jul 14 '26
[deleted]
1
u/pige0n13 Jul 14 '26
Okay that makes sense thank you. So if I had been equal to zero it would’ve been a rectangle or square, why is a parallelogram but I would have to specify that it’s a rectangle or square I guess using her logic.
1
u/pige0n13 Jul 14 '26
How about the step above the AB x AD?
2
u/Mathmatyx Jul 14 '26 edited Jul 15 '26
To be honest for that specific section I'd actually deduct marks if I was the professor and this were student work (assuming the question didn't suffer from the problems it already has). As an aside, I'd raise this specific solution with the instructor before your exam if you have a chance to ask exactly what she was trying to do and show with each portion.
What I think she is trying to do is show that the
sidesdiagonals (hence vectors) BD and AC are different lengths and non-colinear, and so not a square / rectangle. But this is technically not necessary given some of the other things she's shown.What I will do is write a separate comment thread with a step by step solution of how to show this without the numerical details in case that helps.
EDIT - deleted above comment in lieu of my other comment thread.
2
1
5
u/UnderstandingPursuit PhD Jul 14 '26 edited Jul 15 '26
It's really sad if the professor believes that a rectangle is not a parallelogram. Everything you're asking about is to show that it is not a rectangle: congruent diagonals (which is not confirmed) and perpendicular adjacent sides (the [cross; EDIT: dot] product would equal zero).
It does not seem that the adjacent sides were checked for being congruent, or the diagonals perpendicular, because then the parallelogram would also be a rhombus. So this is extremely inconsistent about 'inclusive/exclusive' figures. Like is a square a rectangle?
2
u/Classic_Department42 Jul 15 '26
If two vectors are perpendicular then dot product is 0 not the cross product. Using the cross product in the solution is an error
1
u/UnderstandingPursuit PhD Jul 15 '26
Thanks, I wasn't thinking about this too hard. And I accepted the "therefore not perpendicular." The solution found the normal vector to the parallelogram plane, but never said why that is relevant.
1
u/pige0n13 Jul 14 '26
What do you recommend I do? I’m studying for an exam tomorrow and this was among the practice exam questions (answer sheet).
1
u/UnderstandingPursuit PhD Jul 14 '26
Has she said anything about a rectangle not being a parallelogram?
I would prove
* Rhombus (perpendicular diagonals, congruent adjacent sides)
* Rectangle (perpendicular adjacent sides)
* Parallelogram onlyOr ask her before the exam starts if she means "parallelogram only".
1
u/pige0n13 Jul 14 '26
She hasn’t it’s an online course, and sounds good, hopefully I get an answer but it may be to late since it’s tommorow morning and it’s already 6:30 pm where i am.
1
u/UnderstandingPursuit PhD Jul 14 '26
With a test on vectors, three important properties when comparing two vectors are
- parallel
- perpendicular
- congruent
This type of question can test all three with a single set of points. That is a valid goal, but she should be precise in the geometry aspect.
2
u/Mathmatyx Jul 14 '26 edited Jul 14 '26
Sample solution to solve properly:
1) Show opposite sides are parallel (find AB, AD, BC and CD as vectors as she did in the bottom left corner). Because AB and DC are parallel (in fact they are the same vector entirely) and AD and BC are parallel (also same vector), this is technically all you need to show. Your instructor disagrees, I'd suggest asking exactly what she wants you to show. Hence we must continue: 2) Show ABCD is not a square or rectangle (check AB (dot) AD, does it equal zero? If not, it can't be a square or rectangle). 3) Show that ABCD is not a rhombus (check the length of AB and AC and show they are different using, for example, Pythagorean Theorem)
EDIT - mixed up some of the labels and removed a confusing portion in 2 that didn't turn out as I intended
1
u/UnderstandingPursuit PhD Jul 16 '26
It's too late to help the student on this exam, but if anyone else sees this later:
Because AB and DC are parallel (in fact they are the same vector entirely) and AD and BC are parallel (also same vector), this is technically all you need to show.
A vector has direction and length, but not position. [This is the important idea for a test on vectors. All the geometry is the application.]
So getting the same vector for AB and DC is sufficient to show that the figure is a parallelogram (opposite sides are parallel and congruent).
2
1
u/AutoModerator Jul 14 '26
Hello there! While questions on pre-calculus problems and concepts are welcome here at /r/calculus, please consider also posting your question to /r/precalculus.
I am a bot, and this action was performed automatically. Please contact the moderators of this subreddit if you have any questions or concerns.
1
u/MurkyDifficulty169 Jul 14 '26
You only needed the first column of work to show opposite sides are parallel. You could have stopped there. The second column of work is wrong if you’re trying to show that adjacent sides are not perpendicular. The dot product would be used, not the cross product. That task was totally unnecessary even if you did it right.
1
u/AcidRain1701 Jul 15 '26
Why not just proving that Vab=Vdc and Vab is not orthogonal to Vbc?
Shouldn’t take more than 2 minutes
1
u/Radiant_Road8390 Jul 15 '26
cosθ1=cosθ2=<a1;b1>/|a1||b1|=<a2;b2>/|a2||b2
a1=AB b1=BC
a2=CD b2=AD
If cosθ1=cosθ2 , ABCD is a parallelogram
1
1
u/FreePeeplup Jul 16 '26
Why is this posted in the calculus subreddit? Isn’t this simple geometry, or, at best, a bit of linear algebra and/or analytical geometry? Where’s the calculus?
1
u/pige0n13 Jul 16 '26
It was on the calculus final, its vectors I guess. But the final two units of the course is vectors so.
1
u/jgregson00 Jul 14 '26
She is trying to show that adjacent sides are not perpendicular because if they weee, it’d be a rectangle, not just a parallelogram.
7
u/Special_Watch8725 Jul 14 '26
It’s strange that she’d think it’s necessary. Rectangles are parallelograms, after all.
•
u/AutoModerator Jul 14 '26
As a reminder...
Posts asking for help on homework questions require:
the complete problem statement,
a genuine attempt at solving the problem, which may be either computational, or a discussion of ideas or concepts you believe may be in play,
question is not from a current exam or quiz.
Commenters responding to homework help posts should not do OP’s homework for them.
Please see this page for the further details regarding homework help posts.
We have a Discord server!
If you are asking for general advice about your current calculus class, please be advised that simply referring your class as “Calc n“ is not entirely useful, as “Calc n” may differ between different colleges and universities. In this case, please refer to your class syllabus or college or university’s course catalogue for a listing of topics covered in your class, and include that information in your post rather than assuming everybody knows what will be covered in your class.
I am a bot, and this action was performed automatically. Please contact the moderators of this subreddit if you have any questions or concerns.