So many videos on RUclips just compute this as a given..... it's not intuitive at all that the Det should match the volume but you actually worked it out by hand.... amazing
For anyone watching this now, if know a bit of calc iii or vector algebra, you can prove ad-bc right from the beginning. We know that the area of the parallelogram is the cross product between the two vectors, in this case v1xv2 which equals x. Since the cross product is more intuitively defined in 3d, then we can write x. We get the terms (0-0)-(0-0)+(ad-cb)=ad-bc. Therefore, the determinant of a 2x2 matrix is ad-bc.
Sol, others may have told you this, but I treat your sessions like my DVR. Sometimes I stop you and start over, sometimes I pause you to think about what you have said, sometimes I rewind you a little bit to just confirm what you said and how I thought about it. Also, I sometimes go to other videos that you refer as you go along!
It seems to me that this can also be proven using the cross product. It was earlier shown that the area of a parallelogram is the length of its vectors' cross product, and the cross product of two R3 vectors is the vector (a2b3-a3b2 , a3b1-a1b3, a1b2-a2b1). If you regard two R2 vectors as R3 vectors with zeros for their third elements, then it simplifies to (0 , 0 , a1b2-a2b1).
Some teach forming matrix A by using V1 and V2 as row vectors instead of as column vectors. But the determinant of A is the same. Same for 3x3 matrices: det [A] = det [A transposed].
This is a ridiculously stupid method for computing the area. You could do it in 1/4 the time by subtracting the area outside the parallelogram from the rectangle perpendicular to the coordinate axes that bounds it.
So many videos on RUclips just compute this as a given..... it's not intuitive at all that the Det should match the volume but you actually worked it out by hand.... amazing
For anyone watching this now, if know a bit of calc iii or vector algebra, you can prove ad-bc right from the beginning. We know that the area of the parallelogram is the cross product between the two vectors, in this case v1xv2 which equals x. Since the cross product is more intuitively defined in 3d, then we can write x. We get the terms (0-0)-(0-0)+(ad-cb)=ad-bc. Therefore, the determinant of a 2x2 matrix is ad-bc.
Sol, others may have told you this, but I treat your sessions like my DVR. Sometimes I stop you and start over, sometimes I pause you to think about what you have said, sometimes I rewind you a little bit to just confirm what you said and how I thought about it. Also, I sometimes go to other videos that you refer as you go along!
Thanks. Appreciate your appreciation :)
Thank you very much! Great video.
Excellent video, thanks!
thanks aloooot i learn frrom ur videos much better than books please keep on. espacially on engineringfield.
This is really helpful. Thanks a lot!
It seems to me that this can also be proven using the cross product. It was earlier shown that the area of a parallelogram is the length of its vectors' cross product, and the cross product of two R3 vectors is the vector (a2b3-a3b2 , a3b1-a1b3, a1b2-a2b1). If you regard two R2 vectors as R3 vectors with zeros for their third elements, then it simplifies to (0 , 0 , a1b2-a2b1).
Wow... I never knew how the determinant of a 2x2 matrix is ad - bc. Thanks!
Some teach forming matrix A by using V1 and V2 as row vectors instead of as column vectors. But the determinant of A is the same. Same for 3x3 matrices: det [A] = det [A transposed].
wow you're genius, respect!
if you are interested, proceed to develop volume of parallel-piped and other solids area, volume...using that of determinant directly.
Thanks a lot it's very helpful
But how do you prove the cross product then? Do you simply accept it as given or do you wonder where the algorithm for cross product comes from?
Thank you Alot, for taking some time to do all these videos, i really do appreciate :)
Yes. You do know that u derive the cross product of two vectors by constructing a parallelogram
mind = blown
This seems overly complex. Surely it's just:
1. p = v2.dot(v1)
2. v3 = v1 * p
3. H = v2 - v3
4. Area = v1 * H
do you teach math? if not you should
harry - problem
somewhere else as in?
its is good, but the calculating process is wayyyyy tooo slow! i m assuming ppl here are uni students
i want proof area of the triangle in vector
not problem
lol English
Did not just interpret your comment sexually. No, I did not.
This is a ridiculously stupid method for computing the area. You could do it in 1/4 the time by subtracting the area outside the parallelogram from the rectangle perpendicular to the coordinate axes that bounds it.
This is in the context of linear algebra and will be applicable later. Sometimes Maths is also about proving 1+0=1 but in higher dimension
Very helpful, thanks!