设ab垂直于CD。
它其实是朝垂直于平面的方向。
Well, it's the direction that's perpendicular to this plane.
所以这个外积还是垂直于同一个平面。
梯度向量应该垂直于等值线。
The gradient vector would be going perpendicular to the level.
垂直于梯度的方向上,方向导数为零。
The directional derivative in a direction that's perpendicular to the gradient is basically zero.
我们已经知道，梯度是垂直于等值面的。
We have seen that the gradient is perpendicular to the level surface.
它这样旋转，垂直于圆盘。
And so it was rotating like this and was perpendicular to the disk.
它还是垂直于那些曲线。
现在，垂直于同一个平面的，有两个方向。
Now, there's two different possible directions that are perpendicular to a plane.
或许你可以画出每点都垂直于向量场的曲线。
You might be able to draw curves that are perpendicular to it.
它们平面垂直于重力。
换句话说，它垂直于，两个向量所处的平面。
In other words, it's perpendicular to the plane of the two vectors.
这个力，垂直于斜面。
And then there is the normal force perpendicular to the surface.
所以，我们要怎么找，垂直于这个平面的向量？
So, how do we find this vector that's perpendicular to the plane?
假设我将其绕着，通过这点垂直于黑板，的轴旋转。
And suppose I'm rotating it about an axis perpendicular to the blackboard through that point.
它垂直于曲线。
力是垂直于杆。
所以梯度向量垂直于,很好，这是一个很好的开始。
v So -- So if the gradient vector is perpendicular to v OK, that's a good start.
点P，速度向量,垂直于直线，所以该角的正弦值为。
Well, at point P, the velocity vector QP is perpendicular to the line QP, 1 so the sine of that angle is one.
在南美洲的南端，三座火山垂直于安第斯山脉排列成行。
Near the southern tip of south America, a trio of volcanoes lines up perpendicular to the Andes Mountains.
如果它垂直于这个平面，在那个例子里，它垂直于黑板。
Now, if it's perpendicular to the plane, then in that case, it's perpendicular to the blackboard.
看到了吧，系数变为0了，当直线垂直于法向量的时候。
So, see, this coefficient becomes zero exactly when the line is perpendicular to the normal vector.
也就是，曲线每一点上,垂直于曲线，模长为1的向量。
That means a vector that is at every point of the curve perpendicular to the curve and has length one.
这不过是圆的性质——径向垂直于圆，我来画完这张图吧。
That is a property of the circle that the radial direction is perpendicular to the circle. Actually, let me complete this picture.
那意味着，梯度向量在这点上，垂直于切平面或者是等值面。
Well, that means the gradient is actually perpendicular to the tangent plane or to the surface at this point.
你们有两种选择：，垂直于你们自己，垂直于黑板。
You have two choices: it's either coming at you perpendicular into the blackboard.
为啥呢？呃，就是说，我们知道，怎么画垂直于这个平面的向量。
Why is that? Well, let's say that we know how to find a vector that's perpendicular to our plane.
它移动着，力会，垂直于绳子和r的平面，你们可自己知道为什么。
And as it moves, the torque will always be perpendicular to the plane through the string and r.You can just see that for yourself why that is.
我们是知道如何去求水平集的法向量的，也就是垂直于水平集的法向量。
And we know how to find a normal vector to the level set, namely the gradient vector is always perpendicular to the level set.
我们是知道如何去求水平集的法向量的，也就是垂直于水平集的法向量。
And we know how to find a normal vector to the level set, namely the gradient vector is always perpendicular to the level set.
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