Consider a regular octagon with side length 1, tiled with a finite number of parallelograms. Find the total area of all rectangles among these parallelograms.
(Squares are considered rectangles)
Edit: The point is to find all possible values of the total area.
We are told in various textbooks that the determinant of a matrix formed by three vectors, with the coefficients of the coordinate components making up the rows, can be interpreted as the volume of the parallelepiped formed by the vectors on the same coordinate system (or more technically the spatial distortion on those coordinates).
We are also told that the determinant of the cross product of two vectors is the area of the parallelogram formed by the two vectors joined end-to-end.
The cross product matrix of the two vectors can be interpreted as a matrix having three vectors (the vector <i,j,k>, in addition to the two vectors) joined end to end. The volume of the parallelepiped formed by these three vectors would indeed be equal to the area of the parallelogram, when these two vectors lie on the xy, xz or yz plane (as height would be equal to 1). However, if they do not lie on this plane, the height can vary between zero and root (3).
Hence the volume of the parallelepiped would not be equal to the area of the parallelogram unless the vectors lie on the axes plane. But we are told in calculus classes that we can use the the cross product to find the area of a parallelogram in any case.
Where am I wrong here?
So I googled this and this is the only useful result I got:
Sadly it does not help me though.
The square I have is a child of something and just rotating it on the Y-angle distorts it to said shape. The stated solution didn´t solve it either it still happened.
I feel like this is a common and basic mistake/error but I am unable to fix it. Can anybody help please?
Edit: I did the rotation via
transform.rotation *= Quaternion.AngleAxis(1, Vector3.up);
but it is unimportant since rotating it via the inspector causes the same distortion.
Inspired by this riddle
Consider a regular octagon of side length n. How many ways are there to dissect the octagon into parallelograms with side length 1? You may assume that the vertices of the octagon are labeled (so that you do not need to deal with considering two dissections equivalent if they are the same after rotation).
For a slightly easier variant, use a regular hexagon. For a harder variant, what is the answer for an arbitrary 2m-gon?
Edit: Here is a picture made by /u/JWson of an example when n=1. In this case, every example will be a rotation or reflection of this example, and so there are 8 tilings. In the case of a hexagon, when n=1, there are 2 tilings, which are rotations of each other.
I have a pair of Celestron 15x70 binoculars that were absolutely amazing to look through last night. Standing at the tripod and contorting my neck all sorts of weird directions was not amazing. I'm 95% sure I want a parallelogram mount. More specifically, I want the kind with the L bracket that allows the tripod to be off to the side and lowered down to where the user is sitting comfortably in a chair. Some of the cheaper ones have the mount and tripod in front of the user.
I keep seeing the Orion 05752 Monster Mount & Tripod. That's currently $500 on Amazon. I might as well buy a dob for that price. What rise can you wonderful folks recommend?
One other question: until last night I was planning on spending some of my bonus next spring to get a 10 or 12 inch dob ($1k range), but after last night, I think I'd rather spend a few hundred to make the binoculars more enjoyable. The whole point is to use this stuff, right? I find it more likely that I'll get out binoculars and a chair than a huge scope. Moreover, we have a family membership with our local astronomical society, and once Covid is over, I plan on getting trained on their scopes that I can drive 20 minutes to use any time.
Bottom line: I would rather spend money on binocular equipment that I'll use at home more often and avail myself of astronomy club gear once Covid passes. Is that a good idea?
My teacher had some tech issues so maybe I missed something in his lecture videos videos?Still, I’ve spent two days messing with this thing and I still feel like I’m nowhere close. The deadline was at midnight so I’m just asking now because I feel that I really need to learn how to do this type of problem. It seems super straightforward since vectors B and D should be parallel, right?
A=(-3,5); B=<1,b>; C=(3,6); D=<a,4>
Find values of a and b
(Note: Numbers between arrows are vectors, not Cartesian points)
Is it true a square is a parallelogram?
So I was doing a math assignment and this is really twisting my head.
The area of a parallelogram is Length * Height
The area of a rectangle is side a * side b.
Isn’t a parallelogram just a tilted rectangle?????
Please send help so I can stop thinking about it.
I have a test after new years, and i really need to learn trigonometry but in triangle, trapezoid , rectangle and Parallelogram. Couldnt find any video helpful for these forms, every video is whether circle, or a normal rectangle. any help or a link would be helpful please.
its my college preparation exam, so i guess its all high school level if that makes any sense.
thanks in advance.