Name: 3blue1brown 線性代數精髓系列第 11 章 - 外積在線性變換下的意義（Cross products in the light of linear transformations | Essence of linear algebra chapter 11）
Uploaded: 2021-02-16T09:21:13.000Z
Duration: 13 min 10 s
Description: 3blue1brown 線性代數精髓系列第 11 章 - 外積在線性變換下的意義

I was talking about how to compute a three-dimensional cross product

It's this funny thing where you write a matrix, whose second column has the coordinates of

whose third column has the coordinates of w,

but the entries of that first column, weirdly, are the symbols i-hat, j-hat and k-hat

where you just pretend like those guys are numbers for the sake of computations.

If you just chug along with those computations, ignoring the weirdness,

you get some constant times i-hat + some constant times j-hat + some constant times k-hat.

How specifically you think about computing that determinant

All that really matters here is that you'll end up with three different numbers

that are interpreted as the coordinates of some resulting vector.

From here, students are typically told to just believe that

the resulting vector has the following geometric properties.

Its length equals the area of the parallelogram defined by v and w.

It points in a direction perpendicular to both of v and w.

And this direction obeys the right hand rule

in the sense that if you point your forefinger along v

it'll point in the direction of the new vector.

that you could do to confirm these facts.

But I want to share with you a really elegant line of reasoning.

It leverages a bit of background, though.

So for this video I'm assuming that everybody has watched chapter 5 on the determinant

and chapter 7 where I introduce the idea of duality.

As a quick reminder, the idea of duality is that

anytime you have a linear transformation from some space to the number line,

it's associated with a unique vector in that space

in the sense that performing the linear transformation

is the same as taking a dot product with that vector.

Numerically, this is because one of those transformations

is described by a matrix with just one row

where each column tells you the number that each basis vector lands on.

And multiplying this matrix by some vector v is computationally identical to

taking the dot product between v and the vector you get by turning that matrix on its side.

The takeaway is that whenever you're out in the mathematical wild

and you find a linear transformation to the number line

you will be able to match it to some vector

which is called the “dual vector” of that transformation

so that performing the linear transformation

The cross product gives us a really slick example of this process in action.

It takes some effort, but it's definitely worth it.

What I'm going to do is to define a certain linear transformation from three dimensions

And it will be defined in terms of the two vectors v and w.

Then, when we associate that transformation with its “dual vector” in 3D space

that “dual vector” is going to be the cross product of v and w.

The reason for doing this will be that understanding that transformation

is going to make clear the connection between the computation and the geometry of the cross

remember in two dimensions what it meant to compute the 2D version of the cross product?

you put the coordinates of v as the first column of the matrix

and the coordinates of w is the second column of matrix

There's no nonsense with basis vectors stuck in a matrix or anything like that.

Just an ordinary determinant returning a number.

Geometrically, this gives us the area of a parallelogram

with the possibility of being negative, depending on the orientation of the vectors.

Now, if you didn't already know the 3D cross product

you might imagine that it involves taking three separate 3D vectors u, v and w.

And making their coordinates the columns of a 3x3 matrix

then computing the determinant of that matrix.

geometrically, this would give you the volume of a parallelepiped

depending on the right-hand rule orientation of those three vectors.

Of course, you all know that this is not the 3D cross product.

The actual 3D cross product takes in two vectors and spits out a vector.

It doesn't take in three vectors and spit out a number.

But this idea actually gets us really close to what the real cross product is.

Consider that first vector u to be a variable

What we have then is a function from three dimensions to the number line.

You input some vector x, y, z and you get out a number

by taking the determinant of a matrix whose first column is x, y, z

and whose other two columns are the coordinates of the constant vectors v and w.

Geometrically, the meaning of this function is that

for any input vector x, y, z, you consider the parallelepiped defined by this vector

then you return its volume with the plus or minus sign depending on orientations.

Now, this might feel like kind of a random thing to do.

I mean, where does this function come from?

And I'll admit at this stage of my kind of feel like it's coming out of the blue.

But if you're willing to go along with it

and play around with the properties that this guy has

it's the key to understanding the cross product.

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3blue1brown 線性代數精髓系列第 11 章 - 外積在線性變換下的意義（Cross products in the light of linear transformations | Essence of linear algebra chapter 11）

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