这是平面法向量，和沿直线向量的点积。
It's the dot product between the normal vector of a plane and the vector along the line.
举一个点积的，现实例子。
另一个定义,点积的方法。
跟点积一样-，我会教你们-，两种方法。
I'm going to teach you — - just like with the dot product — two methods.
点积是三个标量的乘积，所以是个标量。
The dot product, being the product of three scalars, is a scalar.
这就是点积。
可以直接用点积来,找到这个角的大小，那么，怎么找呢？
We can just find the angle using dot product So, how would we do that?
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。
Use the vector dot product to find the obtuse angle between two diagonals of a cube .
点积和矩阵。阅读第三章至3.7以及做练习题。
The dot product and matrices. Read Chapter3 through3.7 and do exercises.
矢量乘法有两种,一种叫做“点积“,通常也被叫作“数量积
There are two ways that we multiply vectors and one is called the "dot product" often also called the scalar product.
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。
What we will do is just, at every point along the curve, the dot product between the vector field and the normal vector.
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。
And I can rewrite this in vector form as the gradient dot product the amount by which the position vector has changed.
所要做的功,就等于A到B的,我的力的积分,这是个点积。
So the work that Walter Lewin has to do to bring it from A to B is the integral in going from A to B of my force, dx.
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。
Once you have such a formula, you do the dot product with this vector field, which is not the same as that one.
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 I take the dot product But now, what does it mean that the dot product between OP and a is zero?
因为标量与坐标系无关，故两个矢量的点积称为标量不变量。
Formulae on the base vectors and the coordinate variables derivative going with fluid particle and their application;
提高椭圆曲线点积运算的效率是椭圆曲线研究的一个核心问题。
To improve the efficiency of the algorithm of point multiplication on elliptic curves is a key problem.
此理论的核心内容是传热率和速度矢量与温度梯度的点积值的对应关系。
The core content of the theory is the correspondence between the thermal performance and the integral of the dot product of the velocity and the temperature gradient.
该式不仅含速度矢量与温度梯度矢量的点积项，还含有速度矢量与压力梯度矢量的点积项。
This expression not only contains the term of dot product of velocity vector and temperature gradient, but also contains the dot product of velocity vector and the pressure gradient.
我们来看它的第一个意义，在我们用两个，不一样的向量相乘之前，我们先看一个向量和它本身的点积。
Well, the first thing it means before we multiply two vectors, let's start multiplying a vector with itself That's probably easier.
通过对能量方程进行体积分得到了三维流场中场协同理论的表达式。该式不仅含速度矢量与温度梯度矢量的点积项，还含有速度矢量与压力梯度矢量的点积项。
In this article the mathematical expression of field synergy theory for three dimensional flow with centrifugal force was derived by integrating the energy equation in the concerned volume.
在这一点上西方世界也没有什么不同，只不过积恶更深罢了。
(In this respect Western societies are not that different, just a bit further along the same path).
“好主意，”我回敬他说，“最好留着它，能为你有下位乘客积点德。”
“Great idea,” I replied. “Better save it for your next customer.
同时，日本银行即将为各银行支付盈余公积(surplus reserves)利息——实际是为了赶在利率再次降至零点之前启动量化宽松政策。
Meanwhile, the bank of Japan will start paying interest on Banks' surplus reserves-in effect, allowing quantitative easing to begin before interest rates are again at zero.
“好主意，”我回敬他说，“最好留着它，能为你有下位乘客积点德。”
"Great idea," I replied. "Better save it for your next customer."
鳦之后的四个月，差不多每个月都在下雪，而我就坐在沙发上，看着雪一点点堆积起来。
It snowed almost every day for the next four months, while I sat on the couch and watched it pile up.
我的网站有个“积点”制度，用户可以通过完成任务获得“点数”，比如张贴照片，写博客，收发邮件等。
My website has a point system where the users gain points to do tasks like post pictures, write blogs, send messages, etc.
我的网站有个“积点”制度，用户可以通过完成任务获得“点数”，比如张贴照片，写博客，收发邮件等。
My website has a point system where the users gain points to do tasks like post pictures, write blogs, send messages, etc.
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