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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 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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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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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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Actually, the view of vectors we take nowadays is that vectors are associated with a pair of numbers which, on the rotation of axes, transform like this.
实际上,今天我们讲矢量所用的观点是,矢量是和一组坐标值相对应的,在坐标轴旋转的过程中像这样变换
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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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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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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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That's the very important difference.
这是矢量和标量很重要的区别
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The claim is, I can write any vector you give me as a real scale i plus a real scale j.
也就是说,我可以把任何你给我的矢量,写成 i 和 j 的实系数线性组合的形式
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If you do, the unit vectors are called i prime and j prime, the components of the vector change.
如果这么做,单位矢量就是 i' 和 j',矢量的两个分量也会发生变化
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You can ask yourself, "If you gave me a particular vector, what do I use for Ax and Ay?"
你可以问问自己,"如果给定一个矢量,该怎么确定 Ax 和 Ay"
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There's nothing you can throw at me that I cannot handle with some multiple of i and some multiple of j.
你们找不出任何一个我无法,用 i 和 j 的倍数来表示的矢量
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But since sine square plus cosine square is 1, you'll find this vector has a fixed length R.
但是由于正弦和余弦的平方和为1,你会发现这个矢量模长恒等于 R
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The same vector A can be written either in terms of i and j or in terms of i prime and j prime.
同一矢量 A 可用 i 和 j 表示,也可用 i' 和 j' 表示
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You're asking, "How much i and how much j do I need to build the same object?"
你们会问,"需要多少倍的 i 和多少倍的 j,来构成同一个矢量呢"
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But this is the same vector we are calling i times Ax plus j times Ay.
这和矢量 i ? Ax + j ? Ay 是一样的
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When I give you multiple of i and another multiple of i, there's some has got as its coefficient the sum of the two coefficients.
如果我给你两个不同 i 矢量的倍数,就可以得到某个矢量,且它的系数是这两个系数的和
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You've got to be very used to the notion of taking a vector in some oblique direction and writing it in terms of i and j.
你们应该已经很熟悉,倾斜的坐标系中矢量的概念,和用 i 和 j 来表示该矢量的方法
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In other words, the person wants to ask, "How much i prime and how much j prime do I need to build up the vector A?"
换句话说,有人会问,"我需要用多少倍的 i' 和多少倍的 j',来组成矢量 A"
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So i and j are vectors of length one, pointing along x and y.
和 j 是模长为1的矢量,分别指向 x 轴和 y 轴方向
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First thing you can tell is that if you find the length of this vector, you'll find the square of the x and the square of the y.
首先你可以知道的是,如果你已有了这个矢量的模长,你就可以得到 x 和 y 分量的平方
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i and j are constant, ignore them when you take derivatives.
和 j 是常矢量,求导的时候可以忽略它们
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i and j are unit vectors in the x and y directions.
和 j 分别是 x 和 y 方向上的单位矢量
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i prime and j prime are now rotated unit vectors.
' 和 j' 是旋转后的单位矢量
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If you say, "Add the vectors, " I would just add the x to the x and the y to the y and I'm keeping track of what the sum is.
如果你们说 " 将矢量相加 ",我就把同为 x 轴方向的分量相加,同为 y 轴方向的分量相加,然后就能得出这两个矢量的和是多少了
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Yeah, if you immediately said, "Well, if the vector is , then the vector looks like this.", you're making the assumption that I am writing the vector in terms of i and j.
对,如果你紧接着就说," 如果一个矢量是,那么它就像这样 ",你其实就是假设了,我是按 i 和 j 分量的形式写的这个矢量
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Now, when you work with components, Ax and Ay, if I didn't mention it, they are the components of the vector, you can do all your bookkeeping in terms of Ax and Ay.
当你们在计算分量 Ax 和 Ay 的时候,即使我没有说明,你们也要记得它们是矢量的分量,你们可以都用 Ax 和 Ay 的形式来表示
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This is A and this is B.
这是矢量 A 和矢量 B
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