Multiplication by any other quaternion is completely similar.

与任何他四元数的乘法完全相似。

What I'll show you here is a way to visualize quaternions in their full four-dimensional glory.

我将在这里向您展示一种在完整的四维荣耀中可视化四元数的方法。

Like all other non-unit quaternions, It's invisible to us.

像所有他非单位四元数一样，它对我们来说是不可见的。

On the other hand, all the unit quaternions with negative real part end up somewhere outside that unit sphere.

另一方面，所有具有负实部的单位四元数位于该单位球体外的某个地方。

So the way that this works is slightly more complicated than a single quaternion product.

因此，工作方式比单个四元数乘积稍微复杂一些。

And so far, it's probably not clear how exactly quaternions do describe 3d rotation.

目前为止，我们可能还不清楚四元数如何准确地描述 3d 旋转。

That's right, your phone almost certainly has software running inside it that rely on quaternions.

没错，您的手机几乎肯定运行着依赖四元数的软件。

The 20th century also brought quaternions some more love from a completely different direction, quantum mechanics.

20 世纪也从一个完全不同的方向——量子力学——给四元数带来了更多的爱。

I'll review the method for thinking about quaternion multiplication described last video, explain why half the angle is used, and why you multiply from the right by the inverse.

我将回顾上一个视频中描述的思考四元数乘法的方法，解释为什么使用一半角度，以及为什么从右侧乘以倒数。

Using quaternions to describe 3d rotations is similar, though the look and feel is slightly different.

使用四元数描述 3d 旋转是相似的，但外观和感觉略有不同。

What you do is not just a single quaternion product, but a quaternion sandwich, where you multiply by q from the left and the inverse of q from the right.

您所做的不仅仅是单个四元数乘积，而是一个四元数三明治， 中您从左侧乘以 q， 从右侧乘以 q 的倒数。

The thing is, there are other ways to think about and compute rotations, many of which are way simpler to think about than quaternions.

问题是， 还有他方法来思考和计算旋转，中许多方法比四元数更容易思考。

It's something analogous to the equator of a 3d sphere, and it represents all of the unit quaternions whose real part is 0. What Hamilton would have described as unit vectors.

它类似于 3d 球体的赤道，它代表所有实部为 0 的单位四元数。 汉密尔顿将描述为单位向量。

So after his discovery, he pushed hard for quaternions to be the primary language with which we teach students to describe three-dimensional space, even forming an official quaternion society to proselytize his discovery.

因此，在他的发现后， 他极力推动四元数成为我们教学生描述三维空间的主要语言，甚至组建了一个官方的四元数协会来宣传他的发现。

On the one hand, you could just define quaternion multiplication by giving the rules for how i, j, and k multiply together and saying that everything must distribute nicely.

一方面，您可以通过给出 i、j 和 k 如何相乘的规则来定义四元数乘法，并规定所有内容必须良好分布。

And multiplying quaternion, q1, by another, q2, has the effect of scaling q2 by the magnitude of q1 followed by a very special type of rotation in four dimensions.

将四元数 q1 与另一个 q2 相乘，具有将 q2 按 q1 的大小缩放的效果，然后进行四维旋转的非常特殊类型。

And Hamilton used a special word for quaternions that had no real part and just ijk components, a word which was previously somewhat foreign in the lingo of math and physics: " vector" .

汉密尔顿使用了一个特殊的词来表示四元数，它没有实部，只有 ijk 分量，这个词以前在数学和物理学的行话中有点陌生：“向量”。

And just as the magnitude of a complex number—its distance from zero— is the square root of the sum of the squares of its components, that same operation gives you the magnitude of a quaternion.

正如复数的大小（距零的距离）是分量平方和的平方根一样，同样的运算也可以得四元数的大小。

So j goes to k, k goes to -j, -j goes to -k, and -k goes to j. This gives us a little table for what the number i does to the other quaternions.

因此，j 转 k， k 转 -j， -j 转 -k， -k 转 j。这为我们提供了一个小表格， 显示 i 对他四元数的作用。

So to set the stage, last video I described quaternions, a certain 4-dimensional number system that the 19th-century versions of Wolverine and the old man from Home Alone called " evil" for how convoluted it seemed at the time.

因此，为了做好准备， 我在上一个视频中描述了四元数，这是一种特定的 4 维数字系统， 19 世纪版本的金刚狼和《小鬼当家》中的老人将称为“邪恶” ，因为它在当时看起来非常复杂。