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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

• 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

• 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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