The Vector Cross Product

Vector multiplication can be tricky, and in fact there are two kinds of vector products. We already learned the dot product, which is a scalar, but there is another way to multiply vectors to get another vector, and it's called the cross product. Let's learn how to do this now!
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You do know how to find them. Nothing is stopping you from watching them earlier than the day before the test.

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Y'all are studying this in uni?

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[@cherinhalechantry8786]

Very clearly explained! Thank you Professor Dave!

[@nhaz652]

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)

[@carultch]

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.

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Excellent presentation with explanations that get right to the point.

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Came for the flat earth wreckage, stayed for my Master's degree.

[@carultch]

Well I guess that's one advantage of flat Earthers existing. Getting you introduced to people like Dave to help you earn your degree.

[@junaidfazlani04]

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

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My Sir explained this topic many times but I couldn't relate, when prof. explained understood very clearly ...Thanks...

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You're a life saver Professor Dave !

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3 years later I'm watching this and it still explains it so easily. Keep it up 😊😊😊😊

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Thank you, professor Dave! English is not my mother tongue but I understand Linear Algebra better than I am learning in my university

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Nice tutorials. It helps me alot.

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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

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[@ConceptualCalculus]

Hi Dave. I love your videos. Since we all went online abruptly, I have been using them a lot in my classes. Thank you.

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Thank you professor dave, I hope i pass my exam later. You are very helpful!!

[@Melpomenex]

did you pass?

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[@lukendorf511]

Great explanation, thank you!

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You're currently my favorite math RUclipsr!

[@kanivakil198]

2:23 right-hand rule

[@avamcdonald4437]

so helpful once again. thanks.

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thank you!! this is informative

[@mrudulbuddhadeo7305]

I really understand the concept ....thanku sir

[@lufunonemakhavhani5097]

tnx Mr Dave u rock 😚

[@bernab]

Why a x b as a result is 19 j positive? I think is -19j

[@giacomodoppiazeta8069]

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

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i such love the introduction of this channel it is so shiny

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Thank you professor!

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Do you have a video like this for addition with the same amount set of numbers?

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Helps a lot! Thanks Professor Dave.

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I learned more from this video than what I learned during my entire degree

[@lucialee1232]

Ty this helped a lot

[@varshinilolla3090]

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⊙ ?

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[@goobygoober5789]

Hi is there anyone that can explain for the first example at 1:20 why it is i - j + k ? and not i+j+k or something like that? How do we know whether it is addition/ subtracting i/j/k ?
for some reason i always had trouble with vector math, so please forgive me if the answer is obvious

[@Jcangel26]

I think it is just always set up as i - j + k. I'm not saying it will always end up that way, but that's how it is set up at least

[@ashutoshkumardwivedi3749]

Its why because in an individual determinant we have a1b2-a2b1

[@abdiladifmohamud5957]

I(-1)^a+b + j(-1)^a+b + k(-1)^a+b.
a and b are position of the values I, j and k.
I is in position 1,1 j is in position 1,2 and k is in position 1,3 and if we solve that:
= I(-1)^2 + j(-1)^3 + k(-1)^4
= I-j+k.
Hope that helps.

[@user-ym2mp4jh2c]

Look up cofactor expansion. I like to think of cofactor expansion by covering the current row and column with my finger(s), putting the rest of the (visible) numbers in the smaller determinant
You can think of the signs as a matrix of only signs where each adjacent sign is alternating.
The row or column you choose for calculating the determinant is what decides the sign of each smaller determinant. The top left is *always* a +. You can continue the pattern infinitely.
[ + - ] 2x2 determinant |1 2| = (Using top row, which is [+ -]): +1*(4) - 2*(3)
[ - +] |3 4| = (Using bottom row, which is [- +]): -3*(2) + 4*(1)
For the top row example equation above:
1 is current #, sign is +, cover 2,3, smaller determinant=4.
2 is current #, sign is -, cover 1,4, smaller determinant=3.
You can also use the "teacher-sponsored memorization" approach which is a*d minus b*c: (multiply the main diagonal) and subtract (multiply the minor diagonal).
|a b| = ad-bc
|c d|
[ + - + ] 3x3 determinant | 1 2 3|
[ - + - ] |4 5 6| = (Using top row, which is [+ - +]): +1*|5 6| -2*|4 6| +3*|4 5|
[ + - + ] |7 8 9| |8 9| |7 9| |7 8|
For the top row example above:
1 is current #, sign is +, cover 1's row and column (1,2,3,4,7), smaller determinant=|5 6|
|8 9|
2 is current #, sign is -, cover 2's row and column (1,2,3,5,8), smaller determinant=|4 6|
|7 9|
3 is current #, sign is +, cover 3's row and column (1,2,3,6,9), smaller determinant=|4 5|
|7 8|
[ + - + - ] 4x4 determinant
[ - + - + ]
[ + - + - ]
[ - + - + ] etc
If I choose the top or bottom row (or left or right column) for a 3x3 determinant, my sign coefficients will be +,-,+.
If I choose the middle row or column for a 3x3 determinant, my sign coefficients will be -,+,-.

[@eivakhanam6292]

@@abdiladifmohamud5957 Oh now it makes sense. Thanks and I appreciate your help. A lot of kids at my age won’t understand any of this at all.

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Thank you!

[@laithdarras6389]

Make sure you put the arrow above to indicate that it's a vector!

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great job

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I like the intro so much.and sama as professor dave

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Thank you. 💯

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This right hand rule is the most bull tip i've ever seen in my while life

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Thank you sir.

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i wish indian schools would have teachers like you!!!

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Thanks man I have exam tomorrow and I understand it

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i have an exam tomorrow. this is a great crash course

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I have mine tomorrow as well

[@alusandrea1501]

This is way easier to understand than memorizing a formula.

[@kjasalewjathan]

The right hand rule remimds me of Poyntings vector

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saving me at 1am the night before my mid semester

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thank you prof dave

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aint no way i came back 3 years later cos I forgot

[@danieljulian4676]

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.

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thank you

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Thanks sir

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Could you do a playlist on dynamics?

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check my classical physics playlist

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Love from India🇮🇳🇮🇳♥️

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State :telangana

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thanks!

[@sarahelana535]

Is the cross product a resulting vector? Or is that term only used when adding vectors?

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yay, naild it

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Love you sir

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Thanks dude

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Well, that's the fastest I understood anything in linear algebra

[@rslitman]

Why isn't the Vector Dot Product video not in the Linear Algebra playlist?

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Thanks

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I love that introduction song

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Tnq u professor dev sir

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what a prof!

[@jaclynrosenthal6939]

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?

[@carultch]

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.

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Me too! I use the voice search and sing "Cross Products of Vectors by Professor Dave explains"

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I wish this man knows how he saves our lives in campus

[@gamexx6676]

why did you subtract at 1:50

[@veerrajuannamdevula9939]

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?

[@ProfessorDaveExplains]

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!

[@veerrajuannamdevula9939]

Sir is the tutorial in linear algebra series?

[@ProfessorDaveExplains]

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

[@veerrajuannamdevula9939]

Professor Dave Explains okay sir I will find it out!!

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the best to ever exist

[@khwajaabdurrehman1375]

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.

[@ProfessorDaveExplains]

Earlier in the big math playlist