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Physicists, air traffic controllers, and video game creators all have at least one thing in common: vectors.
物理学家，航空管制员以及电子游戏设计者都有一个共同的东西：矢量。
What exactly are they, and why do they matter?
它们到底是什么？又为什么这么重要？
To answer, we first need to understand scalars.
为了回答这个问题，首先我们需要理解标量。
A scalar is a quantity with magnitude.
标量是一个有大小的量。
It tells us how much of something there is.
它告诉我们一个东西的多少。
The distance between you and a bench,
你和长椅之间的距离，
and the volume and temperature of the beverage in your cup are all described by scalars.
杯子里饮料的体积和温度，都由标量来描述。
Vector quantities also have a magnitude plus an extra piece of information, direction.
矢量也有大小，此外还有一个另外的信息：方向。
To navigate to your bench, you need to know how far away it is and in what direction,
要走到长椅边上，你得知道它离你多远，还要知道要走的方向，
not just the distance, but the displacement.
不是距离，而是位移。
What makes vectors special and useful in all sorts of fields
使矢量特殊并且在各个领域中都有用的
is that they don't change based on perspective but remain invariant to the coordinate system.
是它们不随视角的变化而变化，而是相对参考系保持不变。
What does that mean? Let's say you and a friend are moving your tent.
这是什么意思呢？比如你和你的朋友要搬一个帐篷。
You stand on opposite sides so you're facing in opposite directions.
你们站在帐篷两边，所以你们面对着相反的方向。
Your friend moves two steps to the right and three steps forward
你的朋友向右走两步，向前走三步，
while you move two steps to the left and three steps back.
而你向左走两步，向后退三步。
But even though it seems like you're moving differently,
即使看起来你们的移动方向不同，
you both end up moving the same distance in the same direction following the same vector.
你们其实都像矢量描述的那样，向相同的方向移动了相同的距离。
No matter which way you face, or what coordinate system you place over the camp ground, the vector doesn't change.
无论你们朝向哪个方向，或是你们在地上建起怎样的坐标系，这个矢量都不变。
Let's use the familiar Cartesian coordinate system with its x and y axes.
我们用很熟悉的、有x和y两条坐标轴的笛卡尔坐标系解释一下。
We call these two directions our coordinate basis because they're used to describe everything we graph.
我们把这两个方向叫做基向量，因为它们可以描述图中的任何向量。
Let's say the tent starts at the origin and ends up over here at point B.
我们设帐篷从原点开始，最后被搬到了B点。
The straight arrow connecting the two points is the vector from the origin to B.
连接两点的直箭头就是从原点到B点的矢量。
When your friend thinks about where he has to move,
当你的朋友思考他该怎么移动时，
it can be written mathematically as 2x + 3y, or, like this, which is called an array.
这个向量可以在数学上被写成2x+3y，或者表示成这样，叫做一个数组。
Since you're facing the other way, your coordinate basis points in opposite directions,
但你正好面对着相反的方向，你的基向量也指向相反的方向，
which we can call x prime and y prime, and your movement can be written like this, or with this array.
叫做x'和y'，你的运动可以被表示成这样，或者用这个数组。
If we look at the two arrays, they're clearly not the same, but an array alone doesn't completely describe a vector.
如果我们看这两个数组，它们显然不同，但是一个数组并不能完全表示一个矢量。
Each needs a basis to give it context,
数组需要基向量来给条件，
and when we properly assign them, we see that they are in fact describing the same vector.
当我们把它们安排好，就可以看出它们其实描述的是同一个矢量。
You can think of elements in the array as individual letters.
你可以把数组中的元素想成一个个字母。
Just as a sequence of letters only becomes a word in the context of a particular language,
就像一串字母在已知语言的条件下才能成为一个单词，
an array acquires meaning as a vector when assigned a coordinate basis.
一个数组在给了基向量的条件下才有了矢量的意义。
And just as different words in two languages can convey the same idea,
就像不同的单词在不同语言里有相同的含义，
different representations from two bases can describe the same vector.
不同基向量下的不同数组也可以表示同一个矢量。
The vector is the essence of what's being communicated, regardless of the language used to describe it.
矢量是交流的精髓，无论用哪组基向量来描述。
It turns out that scalars also share this coordinate invariance property.
标量也有在不同基坐标下不变的性质。
In fact, all quantities with this property are members of a group called tensors.
事实上，所有有这个性质的量都是“张量群”的成员。
Various types of tensors contain different amounts of information.
不同类型的张量包含不同的信息量。
Does that mean there's something that can convey more information than vectors? Absolutely.
这是否意味着还有比矢量表达更多信息的量呢？当然。
Say you're designing a video game, and you want to realistically model how water behaves.
比如你在设计一个电子游戏，你想要真实地模拟水的行为。
Even if you have forces acting in the same direction with the same magnitude,
即使有些力作用方向相同，大小也相同，
depending on how they're oriented, you might see waves or whirls.
由于它们的作用点不同，结果可能是波或者漩涡。
When force, a vector, is combined with another vector that provides orientation,
当一个矢量：力，与另一个描述作用点的矢量结合，
we have the physical quantity called stress, which is an example of a second order tensor.
我们就有了一个叫“应力”的物理量，应力是一个二阶张量的例子。
These tensors are also used outside of video games for all sorts of purposes,
这些张量也被用于电子游戏之外的各种用途，
including scientific simulations, car designs, and brain imaging.
包括科学模拟、汽车设计和脑成像。
Scalars, vectors, and the tensor family present us with a relatively simple way of making sense of complex ideas and interactions,
标量、矢量和张量家族给了我们一个相对简单的方法来解释复杂的动作和物理量，
and as such, they're a prime example of the elegance, beauty, and fundamental usefulness of mathematics.
就这样，它们是数学的优雅和美，以及实用性的最好的例子。
