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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"
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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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