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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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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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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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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 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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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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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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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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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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It does represent the sum, in the sense that if i gave you four bucks and I gave you five bucks, I gave you effectively nine.
它表示的就是这两个矢量的和,好比是如果我给了你四美元,然后再给你五美元,我其实给了你九美元
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When we do relativity, we'll be dealing with vectors in space-time and we'll find that different observers disagree on what is this and what is that.
我们学习相对论的时侯,会涉及到时空矢量的问题,我们会发现观测者们对于观测的结果,有着不同的看法
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So the minus vector is the same vector flipped over, pointing the opposite way.
因此负矢量即为同一矢量的翻转,指向相反的方向
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Okay, so that tells you how to add two vectors and get a sum.
这就是矢量的加法法则
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The idea that he had was, if you go from me to you, with a clockwise rotation, you go from you to me by a counterclockwise rotation.
他的方法是这样的,如果这个矢量从这转到那是顺时针方向,那么这个矢量转回来就是逆时针方向
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So multiplying a vector by a number means stretch it by that factor.
所以用数字乘以矢量,就相当于将它伸长了相应的倍数
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Okay, so the rule for adding the two vectors is, you draw the first one and at the end of that first one, you begin the second one.
矢量的加法法则就是,先画出第一个矢量,然后在它的终点,再画第二个矢量
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Everything is the same as in 1D, except everybody's a vector now.
所有的都和一维下是相同的,除了都换成矢量了
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It doesn't matter the sequence in which you add the two vectors.
在做两个矢量的加法时无所谓相加的顺序
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The tangent's pointing towards the direction you are headed at that instant.
切线的方向就是,速度矢量在那个瞬时的指向
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That is, the vector itself has a life of its own.
也就是说,矢量本身是独立的
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Why is the derivative of a vector also a vector?
为什么矢量的导数还是一个矢量呢
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The length of the vector is the length of the vector.
这个矢量的模长是保持不变的
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The components of the vector are not the same.
矢量的分量和原来的也不再相同
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They depend on who is looking at the vector.
它们的值依赖于矢量的观察者
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I'm going to introduce two very special vectors.
接下来我要介绍两个非常重要的矢量
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So the components of vector are not invariant.
因此矢量的分量不是一成不变的
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Pick any two perpendicular directions Then the same entity, the same arrow which has an existence of its own, independent of axis, can be described by you and me using different numbers.
选取两个互相垂直的方向,这样同样的物体,同一矢量,并且是独立于坐标轴而存在的,可以被你和我用不同的数字来描述
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Even though we started with a single vector, which is the position vector, we're now finding out that its derivative has to be a vector and the derivative of the derivative is also a vector.
即使我们从单个矢量出发,即位移矢量,我们现在也能得出它的导数是矢量,而且导数的导数也是一个矢量
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This is the rule for adding vectors.
这就是矢量的加法法则
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