What is a vector? - David Huynh

譯者: Helen Chang 審譯者: S Sung

Physicists,

物理學家

air traffic controllers,

航管員

and video game creators

和電玩遊戲創作者

all have at least one thing in common:

都至少有個共同點：

vectors.

「向量」

What exactly are they, and why do they matter?

到底什麼是向量，為何重要呢？

To answer, we first need to understand scalars.

先要了解「純量」才能回答這問題

A scalar is a quantity with magnitude.

純量是具有大小的量

It tells us how much of something there is.

告訴我們東西的量有多少

The distance between you and a bench,

你和長凳間的距離

and the volume and temperature of the beverage in your cup

你杯中飲料的體積和溫度

are all described by scalars.

都以純量來描述

Vector quantities also have a magnitude plus an extra piece of information,

向量也有大小

另外加上「方向」這個信息

direction.

To navigate to your bench,

要行至你的長凳

you need to know how far away it is and in what direction,

需要知道它距離多遠 和位在哪個方向

not just the distance, but the displacement.

不只是距離，還有位移

What makes vectors special and useful in all sorts of fields

向量特殊且可用於各領域的原因是：

is that they don't change based on perspective

向量不因視角而改變

but remain invariant to the coordinate system.

而是在「座標系」裡維持不變

What does that mean?

這意味著什麼？

Let's say you and a friend are moving your tent.

假設你和朋友一起移動帳篷

You stand on opposite sides so you're facing in opposite directions.

你們二人面對面站著 所以你們二人面對不同的方向

Your friend moves two steps to the right and three steps forward

你的朋友右移兩步，前進三步

while you move two steps to the left and three steps back.

而你左移兩步，後退三步

But even though it seems like you're moving differently,

儘管似乎你們二人的移動不同

you both end up moving the same distance in the same direction

事實上，你們移動的方向和距離相同

following the same vector.

遵循相同的向量

No matter which way you face,

無論你面向哪個方向

or what coordinate system you place over the camp ground,

或你的營地落腳於哪個座標系

the vector doesn't change.

向量是不變的

Let's use the familiar Cartesian coordinate system

我們熟悉的「笛卡爾座標系」

with its x and y axes.

有 x 軸和 y 軸

We call these two directions our coordinate basis

我們稱這兩個方向為「基礎座標」

because they're used to describe everything we graph.

被用來描述所有的圖形

Let's say the tent starts at the origin and ends up over here at point B.

假設帳篷從原點被搬至 B 點

The straight arrow connecting the two points

則連接原點到 B 點的直箭頭

is the vector from the origin to B.

就是從原點到 B 點的向量

When your friend thinks about where he has to move,

你的朋友可把他想要移動的量

it can be written mathematically as 2x + 3y,

以數學式寫為 2x+3y

or, like this, which is called an array.

或者寫成這樣，稱為「陣列」

Since you're facing the other way,

因為你面對著反方向

your coordinate basis points in opposite directions,

你的基礎座標指向相反的方向

which we can call x prime and y prime,

我們稱為 x’ 和 y’

and your movement can be written like this,

你的移動可以寫為這樣

or with this array.

或這個陣列

If we look at the two arrays, they're clearly not the same,

若單看兩個陣列，它們顯然不同

but an array alone doesn't completely describe a vector.

但單用陣列不足以完全描述向量

Each needs a basis to give it context,

每個陣列都需有基礎座標 來表達前後的關係

and when we properly assign them,

在正確地指派表達後

we see that they are in fact describing the same vector.

我們會看到實際上 二者描述相同的向量

You can think of elements in the array as individual letters.

想像陣列中的元素為個別字母

Just as a sequence of letters only becomes a word

如同排序的字母 需在特定的語言中才成為詞

in the context of a particular language,

an array acquires meaning as a vector when assigned a coordinate basis.

陣列也需依循基礎座標 才有了向量的意義

And just as different words in two languages can convey the same idea,

又像在不同的語言中 可用相異的字傳達相同的想法

different representations from two bases can describe the same vector.

兩個不同的基礎座標 也可描述相同的向量

The vector is the essence of what's being communicated,

向量是溝通的本質

regardless of the language used to describe it.

不論以何種語言描述它

It turns out that scalars also share this coordinate invariance property.

原來純量也具有同樣的座標不變特性

In fact, all quantities with this property are members of a group called tensors.

其實，所有具有這特性的量 都是「張量」的成員

Various types of tensors contain different amounts of information.

不同類型的張量含有不同的信息

Does that mean there's something that can convey more information than vectors?

這豈意味著： 有比向量傳達更多信息的張量？

Absolutely.

絕對有

Say you're designing a video game,

假設你設計一款電玩遊戲

and you want to realistically model how water behaves.

想要真實地模擬水的動作

Even if you have forces acting in the same direction

即使有同方向、同大小的施力

with the same magnitude,

depending on how they're oriented, you might see waves or whirls.

取決於力量的方向 你可能看到波浪或者漩渦

When force, a vector, is combined with another vector that provides orientation,

當某一力量、向量 與另一提供方向的向量結合

we have the physical quantity called stress,

我們得到稱為「應力」的物理量

which is an example of a second order tensor.

這是個「二階」張量的例子

These tensors are also used outside of video games for all sorts of purposes,

除了電玩遊戲外 這些張量也被用於各種目的

including scientific simulations,

包括科學模擬

car designs,

汽車設計

and brain imaging.

和腦成像

Scalars, vectors, and the tensor family present us with a relatively simple way

純量、向量，和張量 提供我們一個相較簡單的方法

of making sense of complex ideas and interactions,

以明瞭複雜的含意，和彼此的互動

and as such, they're a prime example of the elegance, beauty,

因此，它們是數學優雅、俊美

and fundamental usefulness of mathematics.

以及具有實用性的好例子