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!

[@user-de3di8el2i]

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 .

[@prashantsemwal3535]

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 4!......
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 fruits......
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)

[@prashantsemwal3535]

I'll still give it a thumbs up though. -^_^-

[@kashifhussains2951]

Can you answer it plz...
Btw this asserts more to my belief that indians have something injected in them naturally to understand maths more deeply

[@Ozymandi_as]

He's made a video about Geogrbra, not vector algebra or rotational mechanics or electromagnetism or the many other scenarios where the utility of vector cross products has been amply demonstrated! Unfortunately, the way vectors and matrices are often taught does not really give much indication of why they exist or anyone should care. So the various rules, operations and identities seem arbitrary and laborious. But there comes a point when you begin to realize they're everywhere. The determinant of a matrix is 0 if two rows are the same because this would imply two parallel or antparallel vectors, which by definition cannot produce a cross product.

[@gabe-d8u]

hello @prashantsemwal3535 , if you want a good explanation on the cross product, go check out 3Blue1Brown's video called "Cross Product in light of linear transformations" i think that's what its called. Also, the reason why the determinant of a matrix with 2 identical rows is 0 is because one of the vectors is linearly dependent! i suggest you check out 3Blue1Brown's entire series on Linear Algebra my friend in case you want more info, he explains things in a more intuitive way, better than any other teacher who teach this subject that i have met so far

[@juanroldan529]

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

[@MathwithDrew]

Thanks for the kind words! Glad you enjoyed it :)

[@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.

[@justinloiacono6903]

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?"

[@dnclvr]

That basically comes down to convention. Mathematician basically agreed to choose the orientation to be right handed because that is how we work with 3D coordinate system

[@justinloiacono6903]

@@dnclvr thanks, after looking into your response further it began to make sense that our coordinate systems three vector orientation is coming into play. But I might ask, did we pick the orientation of our convention or did our mathematical convention arise out of real world observations?

[@kanhaiyalalrajput3215]

​@@justinloiacono6903 I want to know the same. Like in the equation F=q(v×B), why does the force acts in the direction given by Fleming's left hand rule? If we change our convention, does the direction of force also changes? This confuses me the most.

[@Ozymandi_as]

​@@justinloiacono6903definitely empirically derived from observations of rotational mechanics, electromagnetism, and many other observed physical phenomena.

[@shirsenduroy19-7]

'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

[@uniksalman123]

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

[@brodiedezmend4498]

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

[@APaleDot]

One place to see it is the angle subtraction formula for sine. sin(a - b) = cos(a)sin(b) - cos(b)sin(a). Since cosines are associated with x-coordinates and sines are associated with y-coordinates, we could associate the whole expression with A_x * B_y - B_x * A_y, which is the z-component of the cross-product between A and B. There's a more rigorous connection to made between these things, but that gets the flavor across.

[@kanhaiyalalrajput3215]

They come from similarity between complex numbers and vectors. Also see history of quaternions and how Hamilton discovered them.

[@ModernKira]

Question remains same !!!

[@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!!

[@Bose141]

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 .

[@mansibisht8780]

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

[@i10acious]

Hey mate! I have a doubt in case if you can help me... Why does the cross product of two vectors spit out a vector that's perpendicular to both? like we have a physics equation *vectorF= q( vectorV +vectorB )* which says that Lorentz Force i.e. vectorF is directly proportional to vectorV +vectorB i.e. Velocity of charged particle and magnetic field... I'm just curious to know why Lorentz force is perpendicular to Velocity of charged particle and magnetic field?

[@accio_meatballs]

that's what i wanna know 😭

[@AMANSINGH-tb6pj]

These are the type of doubts that have held me back a lot from learning physics. I have asked many people who are in the field of physics and are very knowledgeable, but have failed to answer intuitively. Many just say that it was figured out experimentalyand then the math was invented to fit the physical outcome. Its very unconvincing, and i highly doubt that its the case.

[@bananighosh7757]

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

[@allisontu7724]

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)!

[@Yusuf-gc5dj]

You explained very well❤

[@arjunchopade3356]

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

[@snehpatel6432]

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

[@APaleDot]

There are actually an infinite line of vectors which are perpendicular to the two input vectors. We determine the length of the output vector based on the area of the parallelogram swept out between the two input vectors, and the direction is determined by the "right-hand rule". To remember the right-hand rule: first do a thumbs-up sign with your right hand, then open your fist a little and imagine pushing the first input vector towards the second with the tips of your fingers. The direction your thumb is pointing while pushing is the direction of the output vector.

[@snehpatel6432]

@@APaleDot but by which consept, we can use 'right hand thumb rule' ?

[@APaleDot]

@@snehpatel6432
The right-hand rule is just a convention to determine which direction the output vector goes. It's the standard convention used by mathematicians, there's nothing inconsistent about using the other direction, mathematicians just decided to use the vector that follows the right-hand rule rather than the left-hand rule.

[@Arya3-fk9wl]

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

[@ccerajgcbhadraclass6003]

Exactly I got your point

[@arijitbanerjee8206]

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

[@krishsinghkshatriya]

Thank you Sir for your efforts

[@smzakirhussain7604]

Excellent!

[@usernamedonald]

origin please

[@bhavesh.adhikari]

its 3pm. glad you saved me! goodnight

[@vvee4725]

for the J portion of the cross product should it be u1v3-u3v1?

[@MathwithDrew]

Nope! The second component of the cross product is the backwards one :)

[@RitikKumar-go7fe]

Nice approach 👍

[@ektube9528]

Why do product is parallel when zero

[@vanhieubui3896]

Interesting!

[@josephfarrell2022]

sorry...there's too much background noise

[@ruhitahmed4148]

Thanks

[@shivangi3030]

Thanks helpful !

[@abhijitroy5375]

Thanks you sir😊👍

[@Rksngeet]

Thanks sir

[@johnmaris1582]

Why? Well it was simply define that way. Geometric definition of cross product in this case make more sense as a n vector is inserted simply because it is define to be so. As for the determinant method are just better algorithm nothing more.

[@PurushottamPatil-fx1rl]

Explain it in geometrical meaning. Don't explain it on paper like you did here using determinant. Give us explanation why moment of force applied on fix point at some distance gives direction perpendicular to plane of force and distance, like in cross product. Explain geometrically so we can understand what is going on inside cross product. Don't explain on paper like this. Use 3D models to explain it.

[@cyrildamoah812]

You did well... try going further next time

[@meetarranjjan]

How to prove two vectors parallel without cross product

[@MathwithDrew]

Two vectors are parallel if they're scalar multiples of each other! No cross product required :)

[@meetarranjjan]

Thanks

[@user-ke9gu5id3r]

Not useful