Perhaps it's clear to you that no matter how your axes are oriented, when you ask, "How long is this arrow?"
这应该是比较显然的,无论坐标轴指向哪里,当你问,"这个矢量有多长"
It comes from the fact that velocity is a vector and you can change your velocity vector by changing your direction.
其原理就是,速度是矢量,你可以通过改变方向来改变速度
But I know when I multiply a vector by a number, I get a vector in the same direction.
但是我所知道的是当矢量乘以一个常数,我会得到一个同方向的矢量
You can imagine that if a vector is viewed from an angle, then its components will vary with the perspective.
你可以想象一下,如果换一个角度观察矢量,其分量会相应地发生变化
It's a simple problem, but I just want to do it so you get used to working with vectors.
这是个简单的问题,但我还是想做一下好让你们习惯矢量运算
The proper way to draw a vector is to draw an arrow that's got a beginning and it's got an end.
画矢量的正确方法是,画一个有起点和终点的箭头
Alright, next thing I want to do is to define the vector that plays the role of the number 0.
接下来我想做的是定义一个矢量,它相当于数字0
This formula doesn't tell you which way it's pointing, because it's a scaler; it's not a vector equation.
这个式子没有说明方向,因为这是一个标量而非矢量方程
The beauty of that is now we have discovered a notion of what it means to multiply a vector by a number.
现在最美妙的就是我们就已经知道,数字乘以矢量的意义
That turns out to be a very nice way to produce vectors, given one vector, the position vector.
这或许是一个得到矢量的很不错的方法,给你一个位置矢量
There's one quantity that's going to come out the same, no matter who is looking at the vector.
但是也有一个量是始终不变的,不管是谁在观察这个矢量
What I did last time was to show you how to handle motion in more than one dimension.
上节课的内容,是用多维矢量描述运动
You can take this vector, multiply it by one number, take that vector, multiply it by another number, add the two of them.
比如将这个矢量,乘上一个数,然后将另一个矢量,乘上另外一个数,把这两个矢量相加
But what we will find is, it's more convenient to lump these two numbers into a single entity, which is called a vector.
但我们会发现,将两个数值看作一个整体会更方便,我们称之为矢量
But it has a property that when you add it to anything, you get the same vector.
但它有个性质,就是用它加任何矢量,你得到的都还是原矢量
The advantage of introducing that guy is that if you like, I can now write an equation for the acceleration as a vector.
引入它的好处是,如果你愿意的话,我可以把加速度写成矢量形式
The sum starts at the beginning of the first and ends at the end of the second.
和矢量就是从第一个矢量的起点,指向第二个矢量的终点
It should have the property that when I add it to anybody, I get the same vector.
它也应该有这种性质,用它加任何矢量,得到的还是原矢量
That limit will be some arrow we can call the velocity at the time and it will always be tangent to the curve.
那个极限也就是一个矢量,我们称之为瞬时速度,并且它总是和轨迹相切的
Again, when you learn the relativity, you will find out there's one vector that's staring at you.
再说一遍,当你学习相对论的时候,你会发现,有一个矢量会始终指向你
If you live in two dimensions or more, you've got to use vectors to describe most things.
如果你要考虑二维或二维以上的问题,大多数情况下需要用矢量来描述
If I just come and say to you, "Here's the vector whose components are 3 and 5.
假如我问你们,"一个矢量的分量分别是 3 和 5
Once you've got that, you can do minus 7 times a vector Just take the vector, multiply it by Pi and flip it over.
明白这点之后,你就可以计算-7乘以矢量,只需用 π 去乘以那个矢量,然后将其方向调转
This is how you get the rules for adding a vector to another vector, then taking a vector and multiplying it by some constant.
以上这些就是矢量加法的法则,以及数乘矢量的法则
You can pick your unit vectors, or what are called basis vectors, any way you like.
你可以任意选取单位矢量,或者也可以叫做基矢量,这都随便你
That's what we're going to talk about a little bit, talk a little bit about vectors.
这就是我们今要讨论的内容,一些关于矢量的内容
If I'm now working in one dimension, it's obvious because I'm not using any vectors.
假设我现在只考虑一维的情形,这很明显,因为我没有用矢量
Most of the time, when we deal with vectors, we don't draw these arrows anymore.
多数情况下,当我们处理矢量时,就不再画出这些箭头了
The most important vector is the position vector that tells you where the object is.
最重要的矢量是位置矢量,用来表示物体的位置
We learned that a vector is a quantity that has a magnitude and a direction.
我们已经知道矢量是一个,既有大小又有方向的量
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