Dot and cross product comparison/intuition | Vectors and spaces | Linear Algebra | Khan Academy

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Dot and Cross Product Comparison/Intuition
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[@MrY3110w]

I'm studying engineering at a "world class" university and you explained better in this ~20 min video than any of my prof did in a week

[@malekibrahim7697]

Why don't you ask your professor questions 1-on-1? I don't ask questions at my high school because I know my teachers don't understand the material well enough to give me reasonable explanations... But with professors, they are quite literally, professionals in their field, so if there is something you don't understand, simply ask them as they will likely be able to give you a good solution.

[@PK-eo8kp]

Where do you go?

[@TomDabektv]

Yep, same same.

[@ice-313]

NTU?

[@chitraraj1]

THANK YOU!!! This was priceless to me. You explained in 10 mins what my professor couldn't in weeks. It's amazing how easy you make it all seem. Thank you sir! College professors should be required to watch your videos before they attempt to teach others!

[@janakruia2782]

This man is responsible for restoring my interest in physics (after it has been sucked out by the "professors" that teach me in the institute), by explaining the logic behind how the things are working!!

[@stjoeshs0912]

I very much appreciate the service you provide here. Too often I've had to settle for simply memorizing a formula or process just to get through questions. You provide clear explanations for the concepts presented by these topics, and that to me is a most valuable thing. You outclass a great many professors in your teaching style.

[@atmjahidhasan8818]

Watching the full playlist in just one night. -_- Learning Linear Algebra in a whole fu*king night! -_-

[@keroleswael9332]

Did you do it?

[@ScorpiaX]

Thank you so much. I have a better intuitive understanding of vectors now... My calc teacher just says the cross product gives you the area of the parallelogram without explaining anything, this definitely clarifies it for me.

[@AlexeyYunoshev]

Thank you so so much! It is so satisfying to understand the intuition behind it. Such a great explanation.

[@carlinberg]

if there only was a shout in skyrim called "Soh Cah Toa" I would never have any problem with trigonometry

[@khyatikejriwal1012]

This video is all a student driven by the question why needs.....thank you!!!!

[@zhannqiya2630]

thanks Sal, I've been confused with the differences of these two things for weeks and your video is the exact answer i need. thanks a lot❤

[@MisterTutor2010]

Speaking of going off on a tangent, the tangent of the angle between the vectors is the ratio of the length of the cross product and the dot product :)

[@tszho6611]

THANKYOU VERY MUCH!! What I want is "understanding" the meaning.... This vid helps alot!

[@dariopy]

@comecra85 It's just defined that way. The concept of cross product was basically "invented" as a vector.

[@WonderlandOverture]

Thank you SO much! This is exactly what I was looking for--your explanation makes so much sense.

[@armageddon.4488]

Finally a video that compares both dot product and cross product, really helpful, thanks a lot :D

[@chewch3518]

Great explanation! This video is so much understandable than any kind of "reference book". Really appreciate this one.

[@vigneshbk5131]

Thanks a lot.. !! I will mark my words here.. if become better in my life I will surely find this channel..!! 😍

[@aeroscience9834]

That was the best. It makes much more sense when you get into the intuition. Great video, subd.

[@dianaduque8438]

this is exactly what i have been looking for, its the perfect way to join the concepts in your brain to something graphic you can analize. Gracias desde méxico!

[@bryxnz]

Vectores? Qué fácil.... Ah, hola, por cierto.

[@User-cc4lz]

you are the best one to make intuition of things

[@Ozarkeree]

"This was a bit of a tangent" No, it was a bit of a cosine! (I'm hilarious)

[@kevinryan4000]

Eureka! Sal you totally allowed me to intuitively see what the cross product was doing when you said that the cross product is a measure of how perpendicular two vectors are as defined by Sin theta.

[@bugfacedog44]

"This was a bit of a tangent" I laughed

[@XinhLe]

The cross product is how they perpendicular to each other, kind of reverse to dot product, cool, thanks.

[@sabinbnjara3174]

I'm very much surprised why were we not taught like this ! we were just asked to remember formulas but not their meanings .this has been absolutely burden in our mind .thanks to khan academy for this video .I am very glad to learn meanings of vectors .thank you.

[@heavydutyloader]

Finally it becomes clear to me. Why explain in such a difficult way, that most teachers do, when you could explain it as easy as it is. Like this?

[@JuteTwine]

That was better than the hot tea with whole milk that I'm about to make. It's the morning of Monday, 6 December, 2021, in cold, damp, windy, cloudy Chicagoland, but I'm feeling so warm and sunny now that Griffiths' _Electrodynamics_ looks much less daunting.

[@HiAdrian]

This is so well done, thanks a lot based Salman.

[@literallyshane4306]

Excellent! 2.04 Bit of a tangent?! :D

[@bakhodirjonkakhkharov344]

You are a gift of God to students)

[@PyMoondra]

Really well done. Thank you.

[@XinhLe]

Cross product is area of the shape formed by two vectors, just awesome.

[@netroware]

"This was a bit of a tangent" -- Oh dear. :-D Sal, you're brilliant.

[@mawuenaackotia2501]

This is an excellent explanation. I now have a better intuition about dot and cross products.

[@diegosjimenez]

I learned alot from this video. thanks, this will help me on my midterm.

[@icoscx]

TY!!!!!!!!! I finally passed my Linear Algebra Exam!

[@zbzb-ic1sr]

I think the geometric definition of the dot product is quite easy to understand. What I can't get a good intuitive feel for is why the matrix multiplication of the components of two arbitrary vectors yield to its dot product.

[@lucasbilik6125]

This is so ridiculously helpful. Thank you very much!

[@ian.ambrose]

This video is very important.

[@ravikiranmusinada2452]

Really Nice stuff & explanation.....Thank you

[@Zakkacer]

Thanks for the awesome video! Very helpful!

[@daviddeng3551]

Very helpful, thanks!!

[@JS-tl5ru]

Thank you made so much more sense!!

[@ScottJ2505]

It seems to me that a simple way to visualize the cross product is a vector, perpendicular to vectors a and b and pointing in the direction dictated by the right hand rule. The magnitude of this perpendicular vector equals the area of a parallelogram whose sides are equal to the magnitude of vectors a and b. So, if vectors a and b are concident (theta equals 0), then the area of the parallelogram,and thus the magnitude of this vector, is zero. If the vectors are perpendicular (theta = 90) then the area and magnitude is a x b. If the angle between vectors a and b is between 0 and 90 degrees, then the area and magnitude is computed as a x b x sin(theta).

[@XinhLe]

Dot product is how much the two things are the same direction. Cos theta equal 0 when the dot product is maximum, the same direction. If they are orthogonal, is 0. Cool, thanks.

[@TOPGBURANDANGO]

i think of knowing the intuition but i come for the actual math because that is the only stuff i actually need

[@strawberrybrowniesforfree]

i have my vectors test tomorrow and im sure ill pass by knowing this :p

[@LAnonHubbard]

This is all very interesting.

[@pussiestroker]

Very intuitive!

[@Reaperking66]

OMG TY, helps so much

[@jaikunjmakwana2836]

Excellent!

[@alicejeon123]

THANK YOU!

[@user-nm7jv2js7z]

So the cross product equals 1. A perpendicular vector to a and b, 2. How much is the vectors a and b perpendicular. 3. The area of a and b. Can someone tell me if am I thinking it right?

[@jithinkuriyedath9917]

Thank you sir!! 😊☺

[@lesliesusil4711]

Why cross product makes a vector and dot product makes a scaler. What was the concept behind the inventor of these product to define these two as a vector and the other one as a scaler. Kindly explain this issue.

[@carultch]

Look up Hamilton and Quaternions. He came up with the concept of quaternions that would work like complex numbers on steroids, and wanted them to replace vectors in general. Instead of writing as a vector in that form, you would write 7*(2/7*i + 3/7*j + 6/7*k). This way, 7 indicates the magnitude, and (2/7*i + 3/7*j + 6/7*k) is a unit quaternion that indicates the direction. The four terms are why they are called quaternions, even though there really are only 3 independent terms and it is a 3-dimensional concept. It didn't catch on as much as Hamilton had hoped, but much of the notation did. The concepts of dot products and cross products that Hamilton had structured for working with quaternions, made it to become operations for vectors in general. Another aspect of quaternions that caught on, is the notation of the i/j/k trio of letters to represent the three Cartesian unit vectors.

[@xToTaLBoReDoMx]

I learned that the length of the vectors were represented by a single line on each side, like an absolute value. Isn't that just what it is? The magnitude of the vector equaling the absolute value of it?

[@carultch]

Some books use double vertical brackets, to specify that it isn't necessarily the same thing as absolute value. I prefer the single vertical brackets, since I don't see the specific difference between magnitude of a vector and absolute value of a real number.

[@jea1080]

brilliant!!!!!

[@MrAshok34]

Brilliant!!!!

[@ImHumblyAmazing]

Wow 10/10

[@Halo276]

Just a question, so we can find the area of the parallelogram as simply just "a" vector cross the "b" vector. Does that mean the area of the parallelogram is also "b" vector cross "a" vector? btw, thank you for all your videos they help ALOT =D

[@carultch]

Yes. However, there is a convention for assigning the direction of area vectors, so it becomes an area vector in the opposite direction when you take the cross product in the other order.

[@stevenydc]

what program are you using in the video?

[@Charlietheleet]

THANKYOUUUUUUUUUUUUUUUUUUUUUUUUUUUU~~~~~~~~~~~~~~~~~

[@DanBrickley]

'bit of a tangent' :)

[@DarthYoungling]

I understand ♥

[@chychychyp]

8:50

[@cyrillemagdi7717]

We have learnt in our maths course that the geometrical meaning of scalar product is is the area of the rectangle, any body has an idea of how that makes sense ???

[@MatthijsRadeis]

That's not the scalar product, but the cross product. As Kahn explains further on in the movie.

[@Judobrinez]

wow

[@abhilashjadoun6076]

hmmm

[@mangore623]

You need to have another look at your math book.

[@thegrumpysock6391]

mangore623 could you inform us why?

[@maheshchandra9164]

Thank you

[@mtndewv]

I may as well be paying my tuition to Sal at this point.

[@xToTaLBoReDoMx]

why did the lines double?

[@myonlynick]

If you don't want all the theory bla bla,skip to the GIST: 12:01 - 13:05 (a.k.a intuition)

[@alkalait]

you are wrong. height is also the opposite of theta. so sin(theta) ||a|| = height. just remember soh cah toa.

[@jonesBONES40]

keep saying hypotenuse haha

[@TomDabektv]

*SOH CAH TOA*