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

仆人把我们包切成圆锥体、圆柱体、平行四边他几种数学图。

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

史蒂芬躺在那里看着大熊。 Elfride 正在考虑一个单调平行四边百叶窗。

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

变换之后 这个正方就变成了我们关心平行四边。

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

这样，具有负 y 坐标向量将对应于该平行四边负。

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

以前，这本书是长方。 现在它更像一个平行四边。

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

VNW叉乘 用X乘法符号表示 就是这个平行四边。

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

比率，小数，还要算平行四边。

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VW 想想它们张成平行四边。

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

它长度等于用vnw定义平行四边。

Look at the parallelogram we defined earlier, which encodes the excordinate of the mystery input vector spanned by that vector.

看看我们之前定义平行四边 它编码了神秘输入向量由该向量张成横坐标。

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， 您应该考虑该平行四边有符号，就像行列式视频所描述那样。

As you apply some matrix transformation, the areas of the parallelograms don't stay the same, they may get scaled up or down.

当您应用一些矩阵变换时，平行四边不会保持不变，它们可能会按比例放大或缩小。

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

这样 一个Y坐标为负向量 就对应了这个平行四边负。

But if V is on the left of W, then the cross product is negative, namely, the negative area of that parallelogram.

但是如果V在W左边 那么叉乘就是负 也就是平行四边负。

Basically, if V is on the right of W, then V cross W is positive and equal to the area of the parallelogram.

基本上 如果V在W右边 那么V叉乘W是正 等于平行四边。

Take the parallelogram defined by the 1st basis factor I had, and the mystery input vector X-Y.

取由第一个基因子定义平行四边 以及未知输入向量X-Y。

Look at that parallelogram we defined early which encodes the x-coordinate of the mystery input vector, spanned by the input vector and j-hat.

看看我们之前定义平行四边， 它编码了神秘输入向量 x 坐标，由输入向量 j-hat 跨越。

So the area of that parallelogram is a sort of screwy, roundabout way to describe the vector's Y cordon.

所以平行四边是一种扭曲 迂回方式来描述矢量Y线。

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.

嗯， 这是我们之前看到平行四边变换版本，是神秘输入向量 y 坐标。

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.

该平行四边是底边 1 乘以垂直于该底边高度，即输入向量 y 坐标。