So, the first things that we learned about in this class were vectors, and how to do dot-product of vectors.
这门课我们首先学习的概念是向量，以及怎样做向量的内积。
You do a search, click on a blue dot to see if a product is near by and then you can check inventory.
你去搜索一个产品，然后点击蓝色的点来看产品是否在当地商店有销售并且检查是否在库。
Once you have such a formula, you do the dot product with this vector field, which is not the same as that one.
一旦你得到一个这样的计算式，你对向量场做点积，这和前面这个不一样。
We can just find the angle using dot product So, how would we do that?
可以直接用点积来,找到这个角的大小，那么，怎么找呢？
OK, now, so there's an interesting thing to note, which is that we can use the usual product rule for derivatives with vector expressions, with dot products or cross products.
还有个很有趣的现象要注意一下，就是我们可以用乘积法则,对向量表达式求导，无论是点乘或叉乘。
These two vectors are perpendicular exactly when their dot product is zero.
当点乘的数量积为零时,这两个向量垂直。
And I can rewrite this in vector form as the gradient dot product the amount by which the position vector has changed.
可以把这些用向量形式重新写下来，就是梯度向量和位置改变量的点积。
The entries in the matrix product are obtained by taking dot products.
矩阵乘积里的元素是通过点乘得到的。
这就是点积。
But now the power that I have to generate is the dot product between the force and the velocity.
但现在我要产生的，功率是力和速度，间的标量积。
What we will do is just, at every point along the curve, the dot product between the vector field and the normal vector.
我们要做的是，沿着曲线的每一点上，取向量场和法向量的点积。
OK, so now if we have a dot product that's zero, that tells us that these two guys are perpendicular.
所以我们现在有了一个点积为，这告诉我们两者是垂直的。
And, if you want to find what goes in a given slot here, then you just look to its left and you look above it, and you do the dot product between these guys.
如果你要找到，这个位置的数是什么，那你就往左，和往上看，然后用它们做点乘。
So, product rule is OK for taking the derivative of a dot product.
那么乘积法则对点乘求导没问题。
Well, we've learned how to detect whether two vectors are perpendicular to each other using dot product.
呃，我们学过怎么用点乘,来判断两个向量是否垂直。
I'm going to teach you — - just like with the dot product — two methods.
跟点积一样-，我会教你们-，两种方法。
There are two ways that we multiply vectors and one is called the "dot product" often also called the scalar product.
矢量乘法有两种,一种叫做“点积“,通常也被叫作“数量积
Now comes my method number two as we had with the dot product, is a geometrical method.
下面是方法,像点乘中一样，是个几何方法。
And, in some cases, for example, if you know that the vector field is tangent to the curve or if a dot product is constant or things like that then that might actually give you a very easy answer.
例如，在某些情况下，如果已知向量场与曲线相切，或者内积是一个常数等等，那么结果将会很简单。
It's the dot product between the normal vector of a plane and the vector along the line.
这是平面法向量，和沿直线向量的点积。
There is another way that you can find the dot product.
另一个定义,点积的方法。
This is different than developing a pure JSF Web application page, which requires the dot (.) in the navigation links. (This requirement may be removed in the future releases of the product.)
这与开发纯的JSFWeb应用程序页面不同，它在导航链接中需要点(.)。
Let us take a down-to-earth example of a dot product.
举一个点积的，现实例子。
If I take the dot product But now, what does it mean that the dot product between OP and a is zero?
如果我用op·a我就得到了，那么OP和A的点积是0代表了什么呢？
If you do the dot product with i hat, you will get the first component 0 that will be x1. One times x1 plus zero, zero.
如果你用i做点乘，你能得到第一个分量,那就是x1，1*x1+0
The dot product, being the product of three scalars, is a scalar.
点积是三个标量的乘积，所以是个标量。
Remove any product labels or Department of Transportation (DOT) shipping hazard labels.
去掉产品标签或交通部(DOT)的运输危险品标志。
Use the vector dot product to find the obtuse Angle between two diagonals of a cube.
用矢量点积求立方体的两条对角线所夹的钝角。
Use the vector dot product to find the obtuse Angle between two diagonals of a cube.
用矢量点积求立方体的两条对角线所夹的钝角。
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