Oooh, now it makes finally sense why the length gives us the area of the Parallelogramm. I just got told that it works, but i didn't see why it makes sense geometrically. The connection was so random for me. Thank you!
As a way to remember this, wouldn't it be nice to point out that component 1 is the determinant of row2+row3? (and similar for other components) :). Of course, this determinant thing will happen again and again in later studies
Hi, As always a brilliant video! But one question arises while watching, why does this only work in R^3? In other words: what makes this space so different from R^2, R^4 or R^42? Thank you in advance!
The one thing that every single video on the cross product, including this one, seems to omit, is how to connect the algebraic formula/definition to a geometric intuition, i.e. *why* (in a geometric sense) does this algebraic definition result in a vector that's perpendicular to *u* and *v* and whose length matches the area of the parallelogram? The same goes for introductions of the determinant: *Why* (in a geometric sense) does the algebraic definition of the determinant of a matrix M give us the volume of the paralellepiped to which which the unit hypercube is mapped by the linear map encoded in M?
@@ChrisOffner I have already some videos about the determinant in German. However, the English ones will be even better because it's embedded in the whole Linear Algebra series then :)
@@brightsideofmaths Right after your video, I watched this video th-cam.com/video/VCHFCXgYdvY/w-d-xo.html and the sound was much louder. Just to give an example. If you can make it louder without much effort, then great. Otherwise, no problem.
It is possible to turn on the captions to read the text rather than listening to it. This is a useful tool to watch these videos even within a noisy environment.
By far one of the best Math channels (and in my opinion recourses in general) out there. Thank you for your efforts. Keep it up
Oooh, now it makes finally sense why the length gives us the area of the Parallelogramm.
I just got told that it works, but i didn't see why it makes sense geometrically. The connection was so random for me.
Thank you!
I will cross my fingers for this video
My favourite channel 😍😍😍
As a way to remember this, wouldn't it be nice to point out that component 1 is the determinant of row2+row3? (and similar for other components) :). Of course, this determinant thing will happen again and again in later studies
Of course, the determinant we will cover soon :)
I don't like this way of explaining it because it makes it look like the cross product can be extended to R^n with n different from 3.
Hi,
As always a brilliant video! But one question arises while watching, why does this only work in R^3? In other words: what makes this space so different from R^2, R^4 or R^42?
Thank you in advance!
We can discuss this in more detail later. However, the orthogonality property cannot be true in R^2 or R^4 as you maybe can visualize.
@@brightsideofmaths I would just like to second this question; I'm also super curious if there are higher dimensional analogues :)
@@nathanwycoff4627 Yeah, there are but not as a product with two factors like here. We will discuss it later :)
@@nathanwycoff4627the only analogue i know of is in 7 dimensions
The one thing that every single video on the cross product, including this one, seems to omit, is how to connect the algebraic formula/definition to a geometric intuition, i.e. *why* (in a geometric sense) does this algebraic definition result in a vector that's perpendicular to *u* and *v* and whose length matches the area of the parallelogram?
The same goes for introductions of the determinant: *Why* (in a geometric sense) does the algebraic definition of the determinant of a matrix M give us the volume of the paralellepiped to which which the unit hypercube is mapped by the linear map encoded in M?
True! I want to cover this in general when we talk about the determinant. However, it felt wrong not to define the cross product early.
@@brightsideofmaths Would be wonderful if you could cover it! Thank you, I greatly enjoy your videos. :)
@@ChrisOffner I have already some videos about the determinant in German. However, the English ones will be even better because it's embedded in the whole Linear Algebra series then :)
@@brightsideofmaths Oh, that would be great. I always wondered why the cross product works :)
Hello, I live in a slightly noisy environment. Can you please increase your mic volume in future videos? I like watching your videos.
This is something TH-cam should do automatically. Is it much quieter than other videos?
@@brightsideofmaths Right after your video, I watched this video th-cam.com/video/VCHFCXgYdvY/w-d-xo.html and the sound was much louder. Just to give an example.
If you can make it louder without much effort, then great. Otherwise, no problem.
@@Independent_Man3 I cannot reproduce this effect. Sorry. Maybe you try another browser?
@@Independent_Man3 Yeah, both videos have exactly the same loudness for my computer.
It is possible to turn on the captions to read the text rather than listening to it. This is a useful tool to watch these videos even within a noisy environment.