Area of a parallelogram

Area of parallelogram using determinants. Why the determinant of a 2x2 matrix is ad-bc. Finally, calculating the volume of a parallelipiped using determinants.

[S]

Thanks again Dr!! I love watching and learning from you. I’m a high school math teacher, and guess what?! I’ll be teaching Calculus next year, which is a dream come true. I’ve always wanted to teach it! I’ll be referencing your channel and blackpenredpen to my students. Thank you for all you do man, I just love it!!!!

[hustlerofculture]

I've been wondering for a while now. I will incorporate this into my explanation of detA.

[madhavpr]

I never really paid attention to the geometrical meaning of determinants and the relationship between volume of parallelopipeds in R^n and the utterly horrible Laplace expansion formula. Today, I just sat down with a notebook, pen and a cup of black coffee, listed the basic properties a determinant function should satisfy
1) Linearity in each column
2) Antisymmetry
3) det(I)= 1
and derived the horrible Laplace expansion formula for the 2x2, 3x3 and the 4x4 case. Turns out, the formula is not mysterious at all. 😀 I think it's not so difficult to extend these to the general nxn case, provided we keep track of the number of switches/permutations of indices.
Finally, I understood why the signs alternate to calculate the determinant via cofactor expansion [+, -, +, - etc] 😀

Oh btw. Great video Dr Peyam !! 😀

[kgregor]

I only know this version: Donaudampfschifffahrtsgesellschaftskapitänswitwe, your version is new to me 😀

And as always very nice video!

[chongli]

Can you relate this to the Jacobian? I think it is so amazing how we can use parallelograms to approximate the area of the image of a square under some mapping and then perform a change of variables to make our integral easier!

[plaustrarius]

Excellent!!!

and we get parallelepipeds for free now basically!

[neilgerace]

This really is a pretty special result ... Can it be generalised to other polygons and polyhedra?

[allyourcode]

This is my favorite explanation of det. I would love to see why this works in higher dimensions. I have seen a good explanation in 3D from Krista King, but what about 4D and higher?

[Chris-yeip]

Can you make a video in your finest Österreich Dialekt?

[newtonnewtonnewton]

Nice video D peyam as usual surly i like it that is the answer of ur quistion in the end of ur video

[hemanthkotagiri]

Can someone name those long words he said in the beginning? I'm a bit curious

[jamiewatt]

Ah so that's why |u . (v x w)| gives the volume of a parallelepiped

[IlTrojo]

Parallelograms are still less interesting than Zur Elektrodynamik bewegter Körper...

[FlyingOctopus]

I transform an a*d size rectangle into parallerogram and b*c sized rectangle. I use the shear operation, which preserves area and some cutting with translation.