Placeholder Image

字幕列表 影片播放

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

    上一個視頻,我已經談到了(點積)內積

  • Showing both the standard introduction to the topic,

    同時給出對這個題材的標準引入

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

    以及更深入的它怎樣和線性轉換有關係。

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

    我想對(叉積)外積也這樣做,

  • which also have a standard introduction

    他也有一個標準的介紹說明

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

    但也可以從線性轉換的角度更深入理解

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

    在這次的影片中,我將分成兩段影片分別說明

  • Here i'll try to hit the main points

    我會盡力集中要點

  • that students are usually shown about the cross product.

    說明學生通常如何被教導十字交乘法

  • And in the next video,

    而在接下來的影片中,

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

    我將展示一個不太常出現在教學中,但學習起來讓人很滿意的觀點

  • it.

    我們從二維空間開始說起

  • We'll start in two dimensions.

    如果你擁有兩個向量,v̅ 和w̅

  • If you have two vectorsand w̅,

    想像他們張開成一個平行四邊形

  • think about the parallelogram that they span out

    我的意思是

  • What i mean by that is,

    如果你複製一個v̅

  • that if you take a copy of

    將他從w̅的尾端移到w̅的頂端

  • and move its tail to the tip of w̅,

    然後再複製一個w̅

  • and you take a copy of

    從v̅ 的尾端移到v̅ 的頂端

  • And move its tail to the tip of v̅,

    四個向量在螢幕上圍成一個平行四邊形

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

    v̅ 和w̅的叉積(外積)

  • The cross product ofand w̅,

    可以用X形狀的乘法符號表示

  • written with the X-shaped multiplication symbol,

    這是平行四邊形的面積

  • is the area of this parallelogram.

    well , 大概是這樣。我們還需要考慮定向

  • Well, almost. We also need to consider

    基本上,如果v̅ 在w̅右邊

  • orientation. Basically, ifis on the

    那v̅ X w̅ 是正值

  • right of w̅, then v̅×w̅ is positive

    而且等於這個平行四邊形的面積

  • and equal to the area of the

    但是,如果v̅ 在w̅左邊

  • parallelogram. But ifis on the left of w̅,

    那麼叉積(外積)為負數

  • then the cross product is negative,

    也就是平行四邊形面積的負值

  • namely the negative area of that

    注意,這表示外積相乘的順序很重要

  • parallelogram. Notice this means that

    如果你交換 v̅ 和 w̅ 的位置

  • order matters. If you swappedand

    改成計算 w̅ X v̅

  • instead taking w̅×v̅, the cross

    那麼 w̅ X v̅ 的叉積(外積)將變成負數

  • product would become the negative of

    不管它之前是怎樣

  • whatever it was before. The way I always

    我總是用這樣的方法來記住這裡的順序

  • remember the ordering here is that when

    當你按照順序求兩個基本向量 î X ĵ 的叉積(外積)時

  • you take the cross product of the two

    計算結果應該是正數。事實上,

  • basis vectors in order, î×ĵ,

    你的基本向量的順序是什麼決定定向

  • the results should be positive. In fact,

    所以因為 î 在 ĵ 右邊

  • the order of your basis vectors is what

    我記得 v̅×w̅ 必須為正數

  • defines orientation so sinceis on

    當 v̅ 在w̅的右邊

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

    所以,以影片所示的向量為例

  • has to be positive wheneveris

    我會跟你說這個平行四邊形的面積等於7

  • on the right of w̅.

    而且因為 v̅ 在w̅左邊

  • So, for example with the vector shown

    叉積(外積)應該是負的,因此v̅ X w̅ 為-7

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

    但是當然地,你想在別人沒有告訴你面積等於多少的前提

  • that parallelogram is 7. And since

    計算出面積等於多少

  • is on the left of w̅, the cross product

    這是行列式上場的時候了

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

    所以,如果你沒有看過這個系列第5章討論行列式的部分

  • But of course you want to be able to

    現在是一個很好的機會去看看

  • compute this without someone telling you

    即使你看過,但已經過了一段時間

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

    我推薦你再看一次

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

    確保這些想法在你的頭腦中仍然新鮮

  • series, where I talk about the

    對於2-維的叉積 v̅×w̅,

  • determinant now would be a really good

    你要做的就是你寫下v̅ 坐標

  • time to go take a look.

    作為矩陣的第一個列

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

    你走取w̅的坐標

  • ago. I'd recommend taking another look

    將它們作爲第二個列然後你

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

    就計算行列式值。

  • For the 2-D cross-product v̅×w̅,

    這是因為一個矩陣其列

  • what you do is you write the coordinates

    表示V和W與對應

  • ofas the first column of the matrix

    移動所述線性變換

  • and you take the coordinates ofand

    基底矢量i和j,以v和w。

  • make them the second column then you

    該決定是所有關於量度

  • just compute the determinant.

    由於轉換型面積怎樣改變。

  • This is because a matrix whose columns

    而且,我們期待原型區域

  • representandcorresponds with a

    在是單位平方擱在i和j。

  • linear transformation that moves the

    改造後,

  • basis vectorsandtoand w̅.

    該廣場被闢為

  • The determinant is all about measuring

    我們關心的平行四邊形。

  • how areas change due to a transformation.

    因此,決定通常這

  • And the prototypical area that we look

    測量由哪些領域的因素

  • at is the unit square resting onand ĵ.

    改變,使這一區域

  • After the transformation,

    平行四邊形;因為它從一個演變

  • that square gets turned into the

    方與面積1日開始。

  • parallelogram that we care about.

    如果v是在W的左邊更重要的是, 它

  • So the determinant which generally

    意味著取向翻轉

  • measures the factor by which areas are

    這種轉變,也就是在

  • changed, gives the area of this

    它裝置,用於行列式為負。

  • parallelogram; since it evolved from a

    舉個例子,讓我們講訴了

  • square that started with area 1.

    坐標負(-3,1)和W具有

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

    坐標為(2,1)。的行列式

  • means that orientation was flipped

    矩陣與這些坐標列

  • during that transformation, which is what

    是(-3·1) - (2·1),

  • it means for the determinant to be negative.

    這是-5。所以看樣子

  • As an example let's sayhas

    我們定義的平行四邊形的面積為5

  • coordinates negative (-3,1) andhas

    並且因為V是W的左邊,它

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

    應該有意義,這個值是

  • matrix with those coordinates as columns

    負。正如你學習任何新的操作

  • is (-3·1) - (2·1),

    我建議你玩弄此

  • which is -5. So evidently the

    在你的頭它的概念只是為了讓

  • area of the parallelogram we define is 5

    那種為了什麼直觀的感受

  • and sinceis on the left of w̅, it

    跨產品的全部。

  • should make sense that this value is

    例如,你可能會注意到,當

  • negative. As with any new operation you learn

    兩個矢量是垂直的或在

  • I'd recommend playing around with this

    至少接近垂直其

  • notion of it in your head just to get

    叉積大於這將是

  • kind of an intuitive feel for what the

    如果他們在非常相似的指向

  • cross product is all about.

    方向。由於該地區

  • For example you might notice that when

    平行四邊形較大時,雙方

  • two vectors are perpendicular or at

    更接近垂直。

  • least close to being perpendicular their

    別的東西,你可能會注意到的是,

  • cross product is larger than it would be

    如果你要擴大其中的一個

  • if they were pointing in very similar

    矢量,或許乘以V通過3

  • directions. Because the area of that

    那麼平行四邊形的面積是

  • parallelogram is larger when the sides

    也是由三個因素放大。

  • are closer to being perpendicular.

    那麼是什麼意思這對於操作

  • Something else you might notice is that

    該3V×寬將正好是三

  • if you were to scale up one of those

    倍V×W的值。

  • vectors, perhaps multiplyingby three

    現在,儘管這一切的是一個

  • then the area of that parallelogram is

    完全正常的數學運算

  • also scaled up by a factor of three.

    我剛才所描述的是技術上不

  • So what this means for the operation is

    交產物。真正的跨產品

  • that 3v̅×w̅ will be exactly three

    是什麼,結合了兩種不同的

  • times the value of v̅×w̅ .

    3D矢量得到一個新的三維矢量。就像 之前,

  • Now, even though all of this is a

    我們仍然要考慮

  • perfectly fine mathematical operation

    由兩個向量定義的平行四邊形

  • what i just described is technically not

    這是一起穿越。和地區

  • the cross-product. The true cross product

    這個平行四邊形的仍然是要

  • is something that combines two different

    發揮很大的作用。為了具體的假設

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

    該區域是2.5的矢量

  • we're still going to consider the

    這裡顯示但正如我所說的交

  • parallelogram defined by the two vectors

    產品是不是一個數字它是一個載體。

  • that were crossing together. And the area

    這個新的向量的長度將是區域

  • of this parallelogram is still going to

    該平行四邊形,在這種情況下

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

    2.5。而這新的方向

  • that the area is 2.5 for the vectors

    矢量將是垂直於

  • shown here but as I said the cross

    平行四邊形。但是,哪一種方式!對不對?

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

    我的意思是有兩種可能的矢量

  • This new vector's length will be the area

    長度2.5的垂直於給定的 平面。

  • of that parallelogram which in this case

    這就是右手法則來

  • is 2.5. And the direction of that new

    英寸把你的右手食指

  • vector is going to be perpendicular to

    在V的方向,然後伸出

  • the parallelogram. But which way!, right?

    中指在W的方向。

  • I mean there are two possible vectors with

    然後,當你點了你的大拇指,這是 該

  • length 2.5 that are perpendicular to a given plane.

    叉積的方向。

  • This is where the right hand rule comes

    例如,讓我們說,五世一個

  • in. Put the fore finger of your right hand

    長度為2指點直矢量

  • in the direction ofthen stick out

    向上在Z方向上,而W是一個載體

  • your middle finger in the direction of w̅.

    與純的Y軸長度2指點

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

    方向。平行四邊形,它們

  • direction of the cross product.

    定義這個簡單的例子是

  • For example let's say thatwas a

    實際上是一個正方形,因為他們是

  • vector with length 2 pointing straight

    垂直,並且具有相同的長度。

  • up in the Z direction andis a vector

    和正方形的面積是4。所以

  • with length 2 pointing in the pure Y

    它們的橫產物應該是一個矢量

  • direction. The parallelogram that they

    與長度4.使用右手

  • define in this simple example is

    通常,他們的跨產品應指向 負X方向。

  • actually a square, since they're

    所以這兩者的積

  • perpendicular and have the same length.

    載體是-4·I。

  • And the area of that square is 4. So

    對於更一般的計算,

  • their cross product should be a vector

    有一個公式,你可以

  • with length 4. Using the right hand

    記住,如果你想要,但它很常見

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

    也更容易記住,而不是一定

  • So the cross product of these two

    過程涉及到3D的決定因素。

  • vectors is -4·î.

    現在,這個過程看起來確實奇怪

  • For more general computations,

    第一。你寫下一個三維矩陣,其中

  • there is a formula that you could

    在第二和第三列包含

  • memorize if you wanted but it's common

    坐標V和W的。但對於

  • and easier to instead remember a certain

    第一列你寫的基向量

  • process involving the 3D determinant.

    I,J和k。然後你計算

  • Now, this process looks truly strange at

    這個矩陣的行列式。該

  • first. You write down a 3D matrix where

    愚蠢可能是在這裡清楚。

  • the second and third columns contain the

    究竟是什麼意思放在一個

  • coordinates ofand w̅. But for that

    載體作為基體的條目?

  • first column you write the basis vectors

    學生常常被告知,這是

  • î, ĵ and k̂. Then you compute

    只是符號的把戲。當你隨身攜帶

  • the determinant of this matrix. The

    出的計算,就好像I,J和K

  • silliness is probably clear here.

    是數字,那麼你得到一些

  • What on earth does it mean to put in a

    這些基向量的線性組合。

  • vector as the entry of a matrix?

    和矢量

  • Students are often told that this is

    由線性組合,學生定義

  • just a notational trick. When you carry

    被告知只相信,是唯一

  • out the computations as if î, ĵ and

    矢量垂直於V和W,其

  • were numbers, then you get some

    幅度是適當的區域

  • linear combination of those basis vectors.

    平行四邊形,其方向服從

  • And the vector

    右手法則。

  • defined by that linear combination, students

    而且,一定!從某種意義上說,這只是一個

  • are told to just believe, is the unique

    記法把戲。但是有一個理由

  • vector perpendicular toandwhose

    在做。

  • magnitude is the area of the appropriate

    這不只是一個巧合

  • parallelogram and whose direction obeys

    決定再次重要。和

  • the right hand rule.

    把基本向量在這些時隙

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

    不只是做一個隨機的事情。至

  • notational trick. But there is a reason

    明白的地方這一切來源於

  • for doing in.

    它有助於使用二元的想法,

  • It's not just a coincidence that the

    我在過去的視頻介紹。

  • determinant is once again important. And

    這個概念是有點重

  • putting the basis vectors in those slots

    雖然,所以我把它放在一個單獨

  • is not just a random thing to do. To

    後續視頻任你是誰

  • understand where all of this comes from

    好奇的了解更多。

  • it helps to use the idea of duality that

    可以說,它屬於精髓外

  • I introduced in the last video.

    線性代數。這裡最重要的組成部分

  • This concept is a little bit heavy

    要知道這是什麼叉積

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

    矢量幾何表示。因此,如果

  • follow-on video for any of you who are

    你想跳過下一個視頻,感覺

  • curious to learn more.

    自由。但是,對於那些你們誰是

  • Arguably it falls outside the essence of

    願意去深一點,誰是

  • linear algebra. The important part here

    好奇之間的連接

  • is to know what that cross product

    這種計算和底層

  • vector geometrically represents. So if

    幾何形狀,想法,我就說說

  • you want to skip that next video, feel

    在接下來的視頻或只是一個真

  • free. But for those of you who are

    優雅的一塊數學。

  • 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.

上一個視頻,我已經談到了(點積)內積

字幕與單字

影片操作 你可以在這邊進行「影片」的調整,以及「字幕」的顯示

B1 中級 中文 美國腔 面積 矩陣 線性 計算 長度 方向

3blue1brown 線性代數精髓系列第 10 章 - Cross products 外積/叉積(Cross products | Essence of linear algebra, Chapter 10)

  • 0 1
    tai 發佈於 2021 年 02 月 16 日
影片單字