And since the volume of a parallelepiped (that's a 3D parallelogram) doesn't depend at all on its slant, then no matter how fast you're moving horizontally the same amount of rain will land on top of you each second.

由于四棱柱（即平行四边形对应的立体模型）的体积并不受它倾斜的角度决定，不管你在水平方向上跑得多快，每秒你头顶上都会淋到等量的雨。

The kinds of things you need to learn in seventh grade math, sure, they're hard: ratios, decimals, the area of a parallelogram.

，数，还要算平行四边形的面积。

How do you calculate the square footage of a parallelogram?

如何计算平行四边形的面积？

Stephen lay watching the Great Bear; Elfride was regarding a monotonous parallelogram of window blind.

史蒂芬躺在那看着大熊。 Elfride 正在考虑一个单调的平行四边形百叶窗。

If they would, for example, praise the beauty of a woman, or any other animal, they describe it by rhombs, circles, parallelograms, ellipses, and other geometrical terms, or by words of art drawn from music, needless here to repeat.

如果他们要赞美一个女人或其他动物的美丽，他们会用菱形、圆形、平行四边形、椭圆等几何术语，或者从音乐中提取的专业词汇来描，就不一一列举了。

Sure enough, there's a miniscule gap between the triangles and trapezoids, forming a parallelogram that stretches the entire length of the board and accounts for the missing square.

果然，三角形和梯形之间有一个微的间隙，形成了一个平行四边形，延伸了整个板的长度并弥补了缺失的正方形。

The area of this parallelogram is its base, 1, times the height perpendicular to that base, which is the y-coordinate of our input vector.

个平行四边形的面积是其底边，1， 乘以垂直于该底边的高度，即输入向量的y坐标。

This gives us the area of a parallelogram spanned out by those two vectors, with the possibility of being negative depending on the orientation of the vectors.

就得到了由两个向量张成的平行四边形的面积 根据向量的方向有可能是负的。

When one hundred and fifty passengers who were destined to be its burden, were on board, they stood like a solid parallelogram, without a possibility of moving; and they were up to their waists in water.

当一百五十名注定要成为它的负担的乘客上船时，他们像一个实心的平行四边形一样站着，没有移动的可能;他们在水没到腰部。

The servants cut our bread into cones, cylinders, parallelograms, and several other mathematical figures.

仆人们把我们的面包切成圆锥、圆柱、平行四边形和其他几种数学图形。

Well, this is the transformed version of that parallelogram we were looking at earlier, whose area was the y-coordinate of the mystery input vector.

好吧， 是之前我们看到的那个平行四边形的变换版本，其面积就是神秘输入向量的y坐标。

Where before you had a nice rectangular-looking book, now you have more of a … parallelogram.

以前，本书是长方形的。 现在它更像一个平行四边形。

For example, if you look the parallelogram spanned by the vector where your first basis vector lands, which is the first column of the matrix, and the transformed version of [x; y], what is its area?

例如，如果你查看由基向量的第一个落点（矩阵的第一列）和[x; y]向量的变换版本所张成的平行四边形，它的面积是多少？

The cross product of VNW, written with the X shaped multiplication symbol, is the area of this parallelogram.

VNW的叉乘 用X形乘法符号表示 就是个平行四边形的面积。

We'll start in two dimensions. If you have two vectors V and W, think about the parallelogram that they span out.

我们从二维开始 如果有两个向量V和W 想想它们张成的平行四边形。

Actually, to be more accurate, you should think of the signed area of this parallelogram, in the sense described by the determinant video.

实际上， 为了更准确地理解， 你应该考虑个平行四边形的有符号面积，就像行列式视频中描的那样。

After the transformation, that square gets turned into the parallelogram that we care about.

变换之后 个正方形就变成了我们关心的平行四边形。

Its length equals the area of the parallelogram defined by vn W.

它的长度等于用vnw定义的平行四边形的面积。

That way, a vector with negative y-coordinate would correspond to a negative area for this parallelogram.

样，一个具有负y坐标的向量就会对应到个平行四边形的一个负面积。

Symmetric to what we did before, you can compute the area of that output parallelogram by creating a new matrix whose first column is the output vector, and whose second column is the same as the original matrix.

与我们之前所做的对称， 你可以通过创建一个新的矩阵来计算输出平行四边形的面积， 个新矩阵的第一列是输出向量，第二列与原矩阵相同。