Hey professor Dave! you are my hero and I would love to be as cool as you someday. It tickles me the way you combat flat eathers while also helping me learn the material my engineering professors can not. You have made my school life much better since i have stumbled upon your videos and i can not thank you enough. Please keep up what you are doing and know that you are a more than a professor you are a HERO!
Hello. I have a qn on Vector Cross Product. 1) Is there a reason why we have to subtract first followed by adding the next vector? 2) Is there an order as to whether or not I should take -5 x 3 first, and then 7 x 4? 3) May I confirm that in each of the vector after solving its multiplication, I have to subtract it? For example, (-5x3) - (7x4)
Multiplication of ordinary numbers is commutative, but this is not the case for the cross product. For the cross product, a cross b is not the same thing as b cross a, because a cross b is the negative of b cross a. You can multiply the individual numbers in any order you want, that carry out the cross product. But to set up the multiplication, you need to keep the vectors in the correct order.
1) yes firstly we took + sign because of the position of that i cap i.e; a11 (1+1=2)(even) so positive then j cap belongs to a12 (1+2=3)(odd) so negative and so on... 2)That's the way of doing determinants 3)that's the whole way of solving determinants
I had been struggling for several months with cross products. I never thought your explanation could help me clear the topics smoothly! Thank you so much professor dave
i just want to learn vector cross product to find area of triangle but as i opened your channel i found ocean of knowledge .omg im shocked...thnak you so much prof deck from this learner from nepal
Cross product: (1) Only in 3D space (2) the value of a determinant is defined to be a scalar quantity, not a vector; the rubric for the cross product is only a mnemonic. Do not study only with this channel unless you wish to treat it simply as prep for exams that will include only very rudimentary "calculate this and give us a number" exercises. In other words, you're not learning linear algebra from Professor Dave in any depth. It's quick, thorough prep for next day's exam if you're only asked to calculate.
Resolve b into two perpendicular components, one parallel to a, and one perpendicular to a; then: magnitude of cross product = [magnitude of a] X [magnitude of the component of b which is at right angles to a (which is "b sin theta") ]
2:00 vector : CROSS product (vector. like finding determinants) . not DOT product (scalar). 2:55 direction of cross product. a x a = 0. 4:00 magnitude / length of vector. parallel vector cross product = 0. 4:154:40 . 5:25 cross product only follows distributive property.
Yes.. parentheses first, so if you add vectors b and c the result is one vector (b+c), therefore you can find the cross product between the two vectors a and (b+c)😊
I LOVE EDUCATION RUclips THANK YOU SO MUCH PROFESSOR DAVE, YOU ARE A STEPPING STONE TO MY CAREER AS AN AEROSPACE ENGINEER. Sorry I really love science.
I do have a question about this part 1:51. How come the answers of 4x7, 4x2, and 3x2 are all negative? Is it always negative when in fact they are all positive?
The second coordinate (j) is the negative of the multiplication. Basically, once you have your result, you just negate the second coordinate. Check 03:22
True. The input vectors don't necessarily need to be perpendicular. When they are not perpendicular, the output cross product vector will still be perpendicular to both of them, and in a direction determined by the right hand rule. Simply let your middle finger rotate to a position other than perpendicular to your index finger. Your thumb will still identify the direction of the cross product resultant. The magnitude of the cross product will be the area of a parallelogram, defined by the two vectors as two of the adjacent sides of the shape. You can use this as a shortcut for finding the area of a triangle among three points in space in general. Find a vector between point A and point B, and another vector between point A and point C. Take the cross product, find its magnitude, and divide by two.
If cross product of two vectors is orthogonal to both of them doesn't that mean new dimension is added? so if there exists two vectors in 2-d plane if we apply right hand rule does the thumb point upward introducing new 3rd dimension Z-axis?
Yes. A cross product of two vectors in 2-d space, requires a third dimension for the resultant. If you cross unit vectors i-hat and j-hat, in that order, the resultant is k-hat, which produces the z-axis. By convention, a standard right-handed coordinate system, puts the x-axis (i-hat unit vector) on the pointer finger, the y-axis (j-hat unit vector) on the middle finger, and the z-axis (k-hat unit vector) on the thumb. The cross product in a purely 2-dimensional space, is just a scalar, rather than a vector. If we lived in flatland, applications of the cross product like torque, would just be scalars.
it's just a dumb thing i made up! if you go to my "just for fun" playlist there is a five hour loop of it in there, just in case you fell like listening for longer!
I don't. I have a different way of remembering how to do the cross product. I imagine a copy of the matrix to the left, and a copy to the right. I then multiply along the down/right positive diagonals, and then along the down/left negative diagonals. Add up the products along positive diagonals. Then subtract the products along the negative diagonals. You get the same answer, and you don't need to think about a negative sign on the j-term. As for in general, it comes from Matrix cofactor expansion, as part of the procedure for finding determinants through sub-determinants. If there were a 4th term, to take a 4x4 matrix's determinant, there would be a negative sign assigned to the 4th term as well. There is a checkerboard of positive and negative signs that applies for determinants in general, when calculating them through sub-determinants. My method of positive and negative diagonals, only works for 2x2 and 3x3 matrix determinants. But that is all we'll ever need to do with cross products in 3D space.
Sir I have a doubt about the multiplication of vectors, I have learnt that cross product is only applicable to 3-D plane vectors so does that mean dot product is only suitable to 2-D vectors?
The dot-product is suitable to any number of dimensions. It just means multiply corresponding components and add them up. The cross product can technically work in 2-dimensions, but it doesn't produce a vector in the 2-dimensional space. Instead, applications of a cross product in 2-dimensions, would just be scalars. 3-dimensional space is the end-of-the-line for where the cross product works, the same way you learn about it in high school level mathematics. There are ways to do the equivalent of a cross product in higher dimensions, but they are beyond my understanding. There are some special cases, like 7-dimensional space, where a cross product still works.
Sir we generally take anticlockwise angles as positive and clockwise as negative right!!Then what if a ray in the x-y plane(endpoint is origin)rotated to Y-Z plane is the angle clock or anti clock?Do we have any math related to this?Are there any tutorials of urs related to this?
that's correct! hmm although i'm not sure what happens in other coordinate planes, i have a tutorial on 3D coordinates that shows which octants are positive and negative relative to all three axes take a look at that and see if it helps!
If you are wondering why of all possible letters, we choose i-hat, j-hat, and k-hat to represent the unit vectors of the x/y/z axes respectively, you are not alone. Some books alternatively opt to use x-hat, y-hat, and z-hat. The reason why i, j, and k are used in this context, has to do with quaternions. William Rowan Hamilton discovered the concept of quaternions and saw their value when applying them to the study of machine dynamics. He anticipated applying them to vector analysis in general, but the concept never really caught on with other mathematicians and scientists. The notation still did catch on, and as a result, there are plenty of textbooks that use i, j, and k with hats to represent unit vectors of the coordinate axes. Quaternions are an extension to the concept of imaginary and complex numbers. In imaginary and complex numbers, we use i to represent the square root of -1, and build additive combinations of a real component not involving i, and an imaginary component that is multiplied by i. With quaternions, we have a unit term that identifies each of the three coordinate axes, that have properties in common with imaginary numbers. Most notably, when you square any one of the unit quaternions, you end up with -1. As a result, we use the letter i's alphabet neighbors as names for the quaternion terms.
Why can't my university prof just put up these videos and call it a day. That way I won't be binge-watching these videos the day before the test
You do know how to find them. Nothing is stopping you from watching them earlier than the day before the test.
Y'all are studying this in uni?
@@yugagalaxa98 Yeah!
@@ConceptualCalculus Woah man you might have a point! Thanks for pointing that out idk what I'd do without you!
@@chenchoon8751 oh. We study it in 11th grade...
Hey professor Dave! you are my hero and I would love to be as cool as you someday. It tickles me the way you combat flat eathers while also helping me learn the material my engineering professors can not. You have made my school life much better since i have stumbled upon your videos and i can not thank you enough. Please keep up what you are doing and know that you are a more than a professor you are a HERO!
I would completely agree....i feel my highschool year has been easier since i found u❤️
this man is single handedly saving my engineering career 😭
us 😭
Omg samee here 😭🥲
@@Sam-em9zy im not a uni student, but the way everyone in these comments sections describe uni, Im cooked
Him and The Chemistry Tutor guy🔥
same bro😂😂
Hello. I have a qn on Vector Cross Product.
1) Is there a reason why we have to subtract first followed by adding the next vector?
2) Is there an order as to whether or not I should take -5 x 3 first, and then 7 x 4?
3) May I confirm that in each of the vector after solving its multiplication, I have to subtract it? For example, (-5x3) - (7x4)
Multiplication of ordinary numbers is commutative, but this is not the case for the cross product. For the cross product, a cross b is not the same thing as b cross a, because a cross b is the negative of b cross a.
You can multiply the individual numbers in any order you want, that carry out the cross product. But to set up the multiplication, you need to keep the vectors in the correct order.
1) yes firstly we took + sign because of the position of that i cap i.e; a11 (1+1=2)(even) so positive then j cap belongs to a12 (1+2=3)(odd) so negative and so on...
2)That's the way of doing determinants
3)that's the whole way of solving determinants
My Sir explained this topic many times but I couldn't relate, when prof. explained understood very clearly ...Thanks...
Mate, thank you so much for these videos, I wouldn't have been able to pass my midterms without you.
I've come across your videos just recently and am so happy I did; thank you SO MUCH for you crystal clear understanding of these concepts!!
I had been struggling for several months with cross products.
I never thought your explanation could help me clear the topics smoothly!
Thank you so much professor dave
i just want to learn vector cross product to find area of triangle but as i opened your channel i found ocean of knowledge .omg im shocked...thnak you so much prof deck from this learner from nepal
You're a life saver Professor Dave !
thank you very much I have final exam tomorrow and your video has rescued me you are the best and I wish you will be good in the future ❤❤
Very clearly explained! Thank you Professor Dave!
Thanks prof, I passed the physics paper with ur help.....ur tutorials are awesome
I used to watch him as I studied for ap's now I watch him at Stanford. thank you goat
Thank you so much sir your explainations are superb and works like a one shot before exams.... By the way, love from India ❤️
Thank you, professor Dave! English is not my mother tongue but I understand Linear Algebra better than I am learning in my university
Cross product: (1) Only in 3D space (2) the value of a determinant is defined to be a scalar quantity, not a vector; the rubric for the cross product is only a mnemonic. Do not study only with this channel unless you wish to treat it simply as prep for exams that will include only very rudimentary "calculate this and give us a number" exercises. In other words, you're not learning linear algebra from Professor Dave in any depth. It's quick, thorough prep for next day's exam if you're only asked to calculate.
3 years later I'm watching this and it still explains it so easily. Keep it up 😊😊😊😊
professor Dave is just the best. Now are understand more about the topic
Nice tutorials. It helps me alot.
@@RajKapoor-ix4mk sum indian ned pusi
Thank you sir☺️☺️
Love from India🇮🇳🇮🇳♥️
Bobs and vegan (ʘᴗʘ✿)
I am from India....
State :telangana
You're currently my favorite math RUclipsr!
PROFESSOR YOU ARE TRULY THE BEST. LOVE YOU, FROM KENYA
i such love the introduction of this channel it is so shiny
because of ur teaching i got excellent marks in this chapter thank you sir thank u very much......
2:23 right-hand rule
Thank you professor dave, I hope i pass my exam later. You are very helpful!!
did you pass?
Thanks sir ur vedios are short and very helpful🙂🙂🙂
This man is saving LIVES 🙌🏻
Wow🎉🎉 I really love your way to explains, it's so easy to understand clearly. Thank you so much
Excellent presentation with explanations that get right to the point.
Resolve b into two perpendicular components, one parallel to a, and one perpendicular to a; then:
magnitude of cross product =
[magnitude of a] X [magnitude of the component of b which is at right angles to a (which is "b sin theta") ]
dude u are greatt you are freaking great
i wish indian schools would have teachers like you!!!
You are the best in the mond bro
2:00 vector : CROSS product (vector. like finding determinants) . not DOT product (scalar).
2:55 direction of cross product.
a x a = 0. 4:00 magnitude / length of vector. parallel vector cross product = 0. 4:15 4:40 .
5:25 cross product only follows distributive property.
Thanks for the video!
In comprehension,
a×b= -3i + 19j + 10k
So,
|a×b| = √(-3)^2 i + (19)^2 j + (10)^2 k = √470
Is √470 = |a| |b| sin⊙ ?
at 5:37 are a,b and c all vectors (in the distributive property)?
Yes.. parentheses first, so if you add vectors b and c the result is one vector (b+c), therefore you can find the cross product between the two vectors a and (b+c)😊
U r the best professor Dave, thanks
bro is science jesus
I LOVE EDUCATION RUclips THANK YOU SO MUCH PROFESSOR DAVE, YOU ARE A STEPPING STONE TO MY CAREER AS AN AEROSPACE ENGINEER. Sorry I really love science.
I do have a question about this part 1:51. How come the answers of 4x7, 4x2, and 3x2 are all negative? Is it always negative when in fact they are all positive?
That's the way the determinant of a 2*2 matrix is defined: [a b | c d] is ad-bc. So the one that is [3 4 | 7 -5] above yields 3*5-4*7.
Great sir, i understood concept very well . Thanks for being there sir
Could you do a playlist on dynamics?
check my classical physics playlist
Thank u so much sir .....u made me understand so clearly😊😊
Very clear,thank you.
When did the dot product video happen or am I just missing something? I don't need it, but I'm just wondering.
Why a x b as a result is 19 j positive? I think is -19j
The second coordinate (j) is the negative of the multiplication. Basically, once you have your result, you just negate the second coordinate.
Check 03:22
Thank god i got this video... thank you sir....
you explain so well! thank you!
thank you!! this is informative
i have an exam tomorrow. this is a great crash course
I have mine tomorrow as well
Great explanation, thank you!
I really understand the concept ....thanku sir
tnx Mr Dave u rock 😚
Thank you professor jave
Came for the flat earth wreckage, stayed for my Master's degree.
Well I guess that's one advantage of flat Earthers existing. Getting you introduced to people like Dave to help you earn your degree.
How to find the direction of the vector product of 2 vectors : 2:25 to 3:05
Hi Dave. I love your videos. Since we all went online abruptly, I have been using them a lot in my classes. Thank you.
it's such a wonderful session thank you, sir!!
Thanks man I have exam tomorrow and I understand it
why did you subtract at 1:50
I learned more from this video than what I learned during my entire degree
thank you prof dave
aint no way i came back 3 years later cos I forgot
Superb sir thank you ❤🙏
Make sure you put the arrow above to indicate that it's a vector!
Is the cross product a resulting vector? Or is that term only used when adding vectors?
I like the intro so much.and sama as professor dave
you're a life saver 🙏🏻
At 2:59, the right right-hand diagram seems to show that a & b are perpendicular to each other, but they don't have to always be perpendicular, right?
True. The input vectors don't necessarily need to be perpendicular. When they are not perpendicular, the output cross product vector will still be perpendicular to both of them, and in a direction determined by the right hand rule. Simply let your middle finger rotate to a position other than perpendicular to your index finger. Your thumb will still identify the direction of the cross product resultant.
The magnitude of the cross product will be the area of a parallelogram, defined by the two vectors as two of the adjacent sides of the shape. You can use this as a shortcut for finding the area of a triangle among three points in space in general. Find a vector between point A and point B, and another vector between point A and point C. Take the cross product, find its magnitude, and divide by two.
This is way easier to understand than memorizing a formula.
My University teaches us the cross product in the form i+j+k not as you have as i-j+k, do they give the same results, do you know why it's different?
I think there is a misunderstanding
When you expand a determinant along any row or column you will always get atleast one negative co factor
Do you have a video like this for addition with the same amount set of numbers?
Why isn't the Vector Dot Product video not in the Linear Algebra playlist?
at 6:00 *a x b* is (27 i + 19 j + 10 k) and not (-3 i + 19 j + 10 k)
When I included the negatives : (-12-15)i-(24-(-5))j+12-(-2))k , I got -27i-19j+14k.
Thank you professor!
you always make me feel physics is easy
saving me at 1am the night before my mid semester
Amazing. Thank you.
Brilliant!!!❤️❤️❤️
So any advices for electrostats ?😅
Professor Dave!!! Every thing is perfect except the audio. Please be loud!!!!
so helpful once again. thanks.
Thank you
The right hand rule remimds me of Poyntings vector
I love that introduction song
If cross product of two vectors is orthogonal to both of them doesn't that mean new dimension is added? so if there exists two vectors in 2-d plane if we apply right hand rule does the thumb point upward introducing new 3rd dimension Z-axis?
Yes. A cross product of two vectors in 2-d space, requires a third dimension for the resultant. If you cross unit vectors i-hat and j-hat, in that order, the resultant is k-hat, which produces the z-axis. By convention, a standard right-handed coordinate system, puts the x-axis (i-hat unit vector) on the pointer finger, the y-axis (j-hat unit vector) on the middle finger, and the z-axis (k-hat unit vector) on the thumb.
The cross product in a purely 2-dimensional space, is just a scalar, rather than a vector. If we lived in flatland, applications of the cross product like torque, would just be scalars.
What is the tune during comprehension...someone plrase tell! I really like it....it is relaxing and satisfying!
it's just a dumb thing i made up! if you go to my "just for fun" playlist there is a five hour loop of it in there, just in case you fell like listening for longer!
Thank you professor!!!
Wait, in which previous video did you discuss vector dot product? I can't seem to find it in any previous video in the linear algebra series.
Earlier in the big math playlist
Well, that's the fastest I understood anything in linear algebra
So helpful, damn. Thank you
why do we use minus for j?
I don't. I have a different way of remembering how to do the cross product. I imagine a copy of the matrix to the left, and a copy to the right. I then multiply along the down/right positive diagonals, and then along the down/left negative diagonals. Add up the products along positive diagonals. Then subtract the products along the negative diagonals. You get the same answer, and you don't need to think about a negative sign on the j-term.
As for in general, it comes from Matrix cofactor expansion, as part of the procedure for finding determinants through sub-determinants. If there were a 4th term, to take a 4x4 matrix's determinant, there would be a negative sign assigned to the 4th term as well. There is a checkerboard of positive and negative signs that applies for determinants in general, when calculating them through sub-determinants.
My method of positive and negative diagonals, only works for 2x2 and 3x3 matrix determinants. But that is all we'll ever need to do with cross products in 3D space.
Thank you!
1:46 shouldnt the determinants for j be (a3*b1)-(a1*b3) not (a1*b3)-(a3*b1)
In that case the negative sign would turn into positive
Thanks :)
The first one shouldn't be -3i+19j-10k?
Yes it should be -27i+19j+10k
yes you are right i find the same things
Sir I have a doubt about the multiplication of vectors, I have learnt that cross product is only applicable to 3-D plane vectors so does that mean dot product is only suitable to 2-D vectors?
The dot-product is suitable to any number of dimensions. It just means multiply corresponding components and add them up. The cross product can technically work in 2-dimensions, but it doesn't produce a vector in the 2-dimensional space. Instead, applications of a cross product in 2-dimensions, would just be scalars.
3-dimensional space is the end-of-the-line for where the cross product works, the same way you learn about it in high school level mathematics. There are ways to do the equivalent of a cross product in higher dimensions, but they are beyond my understanding. There are some special cases, like 7-dimensional space, where a cross product still works.
@@carultch thank you
Sir we generally take anticlockwise angles as positive and clockwise as negative right!!Then what if a ray in the x-y plane(endpoint is origin)rotated to Y-Z plane is the angle clock or anti clock?Do we have any math related to this?Are there any tutorials of urs related to this?
that's correct! hmm although i'm not sure what happens in other coordinate planes, i have a tutorial on 3D coordinates that shows which octants are positive and negative relative to all three axes take a look at that and see if it helps!
Sir is the tutorial in linear algebra series?
i'm not sure which topic i squeezed it in as it's kind of random, but it's definitely in the mathematics playlist, you just have to scroll down a lot
Professor Dave Explains okay sir I will find it out!!
sorry but what does the inclusion of alphabets i, j and k mean? are they there to signify a 3rd row to calculate a 3x3 matrix determinant?
i,j and k represent the different axises of a plane i.e. x-axis = i, y-axis=j and z-axis=k
If you are wondering why of all possible letters, we choose i-hat, j-hat, and k-hat to represent the unit vectors of the x/y/z axes respectively, you are not alone. Some books alternatively opt to use x-hat, y-hat, and z-hat.
The reason why i, j, and k are used in this context, has to do with quaternions. William Rowan Hamilton discovered the concept of quaternions and saw their value when applying them to the study of machine dynamics. He anticipated applying them to vector analysis in general, but the concept never really caught on with other mathematicians and scientists. The notation still did catch on, and as a result, there are plenty of textbooks that use i, j, and k with hats to represent unit vectors of the coordinate axes.
Quaternions are an extension to the concept of imaginary and complex numbers. In imaginary and complex numbers, we use i to represent the square root of -1, and build additive combinations of a real component not involving i, and an imaginary component that is multiplied by i. With quaternions, we have a unit term that identifies each of the three coordinate axes, that have properties in common with imaginary numbers. Most notably, when you square any one of the unit quaternions, you end up with -1. As a result, we use the letter i's alphabet neighbors as names for the quaternion terms.
In 2:24 is the product b*a by putting fingers on b and pointing them toward a .....you said it is a*b
Thank you. 💯