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  • Last video, I've talked about the dot product.

  • Showing both the standard introduction to the topic,

  • as well as a deeper view of how it relates to linear transformations.

  • I'd like to do the same thing for cross products,

  • which also have a standard introduction

  • along with a deeper understanding in the light of linear transformations.

  • But this time I am dividing it into two separate videos.

  • Here i'll try to hit the main points

  • that students are usually shown about the cross product.

  • And in the next video,

  • I'll be showing a view which is less commonly taught, but really satisfying when you learn

  • it.

  • We'll start in two dimensions.

  • If you have two vectorsand w̅,

  • think about the parallelogram that they span out

  • What i mean by that is,

  • that if you take a copy of

  • and move its tail to the tip of w̅,

  • and you take a copy of

  • And move its tail to the tip of v̅,

  • the four vectors now on the screen enclose a certain parallelogram.

  • The cross product ofand w̅,

  • written with the X-shaped multiplication symbol,

  • is the area of this parallelogram.

  • Well, almost. We also need to consider

  • orientation. Basically, ifis on the

  • right of w̅, then v̅×w̅ is positive

  • and equal to the area of the

  • parallelogram. But ifis on the left of w̅,

  • then the cross product is negative,

  • namely the negative area of that

  • parallelogram. Notice this means that

  • order matters. If you swappedand

  • instead taking w̅×v̅, the cross

  • product would become the negative of

  • whatever it was before. The way I always

  • remember the ordering here is that when

  • you take the cross product of the two

  • basis vectors in order, î×ĵ,

  • the results should be positive. In fact,

  • the order of your basis vectors is what

  • defines orientation so sinceis on

  • the right of ĵ, I remember that v̅×w̅

  • has to be positive wheneveris

  • on the right of w̅.

  • So, for example with the vector shown

  • here, I'll just tell you that the area of

  • that parallelogram is 7. And since

  • is on the left of w̅, the cross product

  • should be negative so v̅×w̅ is -7.

  • But of course you want to be able to

  • compute this without someone telling you

  • the area. This is where the determinant comes in.

  • So, if you didn't see Chapter 5 of this

  • series, where I talk about the

  • determinant now would be a really good

  • time to go take a look.

  • Even if you did see it, but it was a while

  • ago. I'd recommend taking another look

  • just to make sure those ideas are fresh in your mind.

  • For the 2-D cross-product v̅×w̅,

  • what you do is you write the coordinates

  • ofas the first column of the matrix

  • and you take the coordinates ofand

  • make them the second column then you

  • just compute the determinant.

  • This is because a matrix whose columns

  • representandcorresponds with a

  • linear transformation that moves the

  • basis vectorsandtoand w̅.

  • The determinant is all about measuring

  • how areas change due to a transformation.

  • And the prototypical area that we look

  • at is the unit square resting onand ĵ.

  • After the transformation,

  • that square gets turned into the

  • parallelogram that we care about.

  • So the determinant which generally

  • measures the factor by which areas are

  • changed, gives the area of this

  • parallelogram; since it evolved from a

  • square that started with area 1.

  • What's more ifis on the left of w̅, it

  • means that orientation was flipped

  • during that transformation, which is what

  • it means for the determinant to be negative.

  • As an example let's sayhas

  • coordinates negative (-3,1) andhas

  • coordinates (2,1). The determinant of the

  • matrix with those coordinates as columns

  • is (-3·1) - (2·1),

  • which is -5. So evidently the

  • area of the parallelogram we define is 5

  • and sinceis on the left of w̅, it

  • should make sense that this value is

  • negative. As with any new operation you learn

  • I'd recommend playing around with this

  • notion of it in your head just to get

  • kind of an intuitive feel for what the

  • cross product is all about.

  • For example you might notice that when

  • two vectors are perpendicular or at

  • least close to being perpendicular their

  • cross product is larger than it would be

  • if they were pointing in very similar

  • directions. Because the area of that

  • parallelogram is larger when the sides

  • are closer to being perpendicular.

  • Something else you might notice is that

  • if you were to scale up one of those

  • vectors, perhaps multiplyingby three

  • then the area of that parallelogram is

  • also scaled up by a factor of three.

  • So what this means for the operation is

  • that 3v̅×w̅ will be exactly three

  • times the value of v̅×w̅ .

  • Now, even though all of this is a

  • perfectly fine mathematical operation

  • what i just described is technically not

  • the cross-product. The true cross product

  • is something that combines two different

  • 3D vectors to get a new 3D vector. Just as before,

  • we're still going to consider the

  • parallelogram defined by the two vectors

  • that were crossing together. And the area

  • of this parallelogram is still going to

  • play a big role. To be concrete let's say

  • that the area is 2.5 for the vectors

  • shown here but as I said the cross

  • product is not a number it's a vector.

  • This new vector's length will be the area

  • of that parallelogram which in this case

  • is 2.5. And the direction of that new

  • vector is going to be perpendicular to

  • the parallelogram. But which way!, right?

  • I mean there are two possible vectors with

  • length 2.5 that are perpendicular to a given plane.

  • This is where the right hand rule comes

  • in. Put the fore finger of your right hand

  • in the direction ofthen stick out

  • your middle finger in the direction of w̅.

  • Then when you point up your thumb, that's the

  • direction of the cross product.

  • For example let's say thatwas a

  • vector with length 2 pointing straight

  • up in the Z direction andis a vector

  • with length 2 pointing in the pure Y

  • direction. The parallelogram that they

  • define in this simple example is

  • actually a square, since they're

  • perpendicular and have the same length.

  • And the area of that square is 4. So

  • their cross product should be a vector

  • with length 4. Using the right hand

  • rule, their cross product should point in the negative X direction.

  • So the cross product of these two

  • vectors is -4·î.

  • For more general computations,

  • there is a formula that you could

  • memorize if you wanted but it's common

  • and easier to instead remember a certain

  • process involving the 3D determinant.

  • Now, this process looks truly strange at

  • first. You write down a 3D matrix where

  • the second and third columns contain the

  • coordinates ofand w̅. But for that

  • first column you write the basis vectors

  • î, ĵ and k̂. Then you compute

  • the determinant of this matrix. The

  • silliness is probably clear here.

  • What on earth does it mean to put in a

  • vector as the entry of a matrix?

  • Students are often told that this is

  • just a notational trick. When you carry

  • out the computations as if î, ĵ and

  • were numbers, then you get some

  • linear combination of those basis vectors.

  • And the vector

  • defined by that linear combination, students

  • are told to just believe, is the unique

  • vector perpendicular toandwhose

  • magnitude is the area of the appropriate

  • parallelogram and whose direction obeys

  • the right hand rule.

  • And, sure!. In some sense this is just a

  • notational trick. But there is a reason

  • for doing in.

  • It's not just a coincidence that the

  • determinant is once again important. And

  • putting the basis vectors in those slots

  • is not just a random thing to do. To

  • understand where all of this comes from

  • it helps to use the idea of duality that

  • I introduced in the last video.

  • This concept is a little bit heavy

  • though, so I'm putting it in a separate

  • follow-on video for any of you who are

  • curious to learn more.

  • Arguably it falls outside the essence of

  • linear algebra. The important part here

  • is to know what that cross product

  • vector geometrically represents. So if

  • you want to skip that next video, feel

  • free. But for those of you who are

  • willing to go a bit deeper and who are

  • curious about the connection between

  • this computation and the underlying

  • geometry, the ideas that I will talk about

  • in the next video or just a really

  • elegant piece of math.

Last video, I've talked about the dot product.

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B1 US cross product vector area perpendicular linear

Cross products | Essence of linear algebra, Chapter 10

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    tai posted on 2021/02/16
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