Why is the cross product PERPENDICULAR to both vectors?

The cross product is useful in multivariable calculus and linear algebra primarily because it lets us find a 3D vector that's perpendicular to two vectors we already have. But one thing we often skip over while talking about it is WHY the cross product ends up being perpendicular to its inputs in the first place. Well, here's one way to justify it!

[姜名孺]

I felt confused about why the cross product is perpendicular to both vectors when I was flipping through my math notes. Thanks a lot for explaining it in a excellent way .

[juanroldan]

Hi man thanks for the contribution. Lineal algebra is so confusing I'm glad there's people explaining it in so simple terms!

[prashantsemwal]

Waaiiiittt a min.... You didn't answer the question at all....
moreover you only added another layer to the complexity... (why is determinant of 3x3 matrix with same row or colum zero?)

it's like a kid asks why is 2x2=4....
and I say since the sqar root of 4 is 2....
squar of 2 is four...
squar of 2 is 2x2...
therefore 2x2 is

I hope you understand...

the question should be tackled by where the formulation of cross product come into place....

for example Addition come into play when we had to know the total number of things in a set of set of things...
2 apple 5 bananas = 7
so we deviced A+B=C...

now where did vector product comes into play.. n why we use cross product will determind the rules required to use to this tool.
(I'll answer the question but I want you to make a video about it... Thanks... n hope your dog is doinng good)

[zpocrm]

I appreciate how you explained the methods for calculating the cross product and dot product. It really clarified things for me, highlighting that the dot product involves projecting vectors, whereas the cross product involves multiplying vectors in 3D space. I also use the concept of torque to understand the cross product better—maximum torque is produced when the applied force is perpendicular to the radius.

[shirsenduroy-]

'The cross product notation u × v was introduced by the American physicist & mathematician J. Willard Gibbs, in a series of unpublished lecture notes for his students atYale University. It appeared in a published work for the first time in the second edition of the book "Vector Analysis", by Edwin Wilson, a student of Gibbs. Gibbs originally referred to u × v as the "Skew Product ". '

source from the book 10th edition ' Elementary Linear Algebra ' by the Author Howard Anton/Chris Rorres

5:05

[mansibisht]

Even if i dont understand you, you are the first one to try.
And none gets anything perfect in first time
Thank u anyway

[justinloiacono]

He did not tell "why" the cross product has the orientation that it does, just how is is computed. Nice explanation of the the dot product defining perpendicular though. Here is my question, "Why does the cross product have a right handed answer, rather than a left handed one?"

[Arya-fkwl]

I am confused with one thing.

When dot product is zero, vectors A and B are perpendicular to each other.

When cross product is zero, vectors A and B are parallel to each other. But I have also read that resulatant of A and B is perpendicular to that plane of A and B.

Please clarify.
Thanks

[McLeo]

I like your videos, but this was not inline with what was mentioned in the title of the video...🤯

[brodiedezmend]

I understand everything in this video, but where does the formula for computing the cross product actually come from?

[Bose]

It had eate my brain for years i destroyed my grades to find it but soon realized i need to focus on grade . I reached a conclusion that angular displacement remains conserved along perpendicular along axis also and we can only define that motion unique with perpendicular only because all objects of same dimensions will have constant angular velocity along axis .

[arijitbanerjee]

Is there same concept of cross product which we apply in algebric product

[bananighosh]

Thank you for explaining it Mathematically, but i still don't understand why angular velocity has to work perpendicular

[allisontu]

Super interesting and great review from 21a! I spy a familiar looking cactus in the background :) hard to believe we were kicked out a full year ago (today)!

[haniamritdas]

It is strange that the criss product is not considered a vector, since it has both a value and a direction determined by the plane that it is orthogonal to. 🤔

[meetarranjjan]

How to prove two vectors parallel without cross product

[arjunchopade]

Hi brother ...nice explanation...I am from India 🇮🇳....where r u from ? ....plz reply brother💚❤️

[Michallote]

Hey man you should really cover Geometric Algebra (Clifford Algebra). The wedge product is so much more intuitive. I really feel the cross product is unintuitive and holds back a great portion of useful physics because it basically interprets something that should be a plane as a vector. And it is in my opinion holding back progress in physics education. The answer to it is to use instead the wedge or outer product, the geometric product and bivectors. Cheers I hope people really look this up if they are confused!!

[ProttoyDas-cmxs]

1:01 but this formula itself is derived using the thumb rule and you r deriving thumb rule using this.Its a kind of loop and makes no sense.

[snehpatel]

We can describe two possibility of vector which perpendicular to both two vector. Which is right and why ?