如何使用矩阵相对于另一个向量旋转参考系？
现实生活中的案例（有助于理解问题）
我正在构建一个可以绕其所有轴（x，y，z）自由旋转的设备，并配备：
- 一个加速度计（A），它给了我指向地球中心的矢量 (Va)
- 提供地球磁场方向的 3D 磁力计 (M) (Vm)
这两个矢量共享相同的参考系 (x ，y，z），但我感兴趣的是找到相对于地球表面指向北方的向量[想象一个手持指南针：我想找到指针应该在哪里指向]。
此视频展示了一个“球形指南针”，其中有我的设备应该以电子方式复制几乎相同的行为。
编码问题
我做了一些研究的 ，在我看来，我应该使用的是 3D旋转矩阵执行以下两个步骤：
- 将Vm的参考系从R旋转到R'，使得y'平行于Va，
- < 不幸的是，我仍然对如何具体
进行操作感到困惑（我以前没有使用向量和矩阵的经验）。让我困惑的一件事是，我可以在谷歌上搜索到的大部分材料都是从角度角度进行讨论的，但是我从两个传感器接收到的数据的形式是 V(Vx, Vy, Vz)，其中 Vz、Vy 和 Vz 是 V 沿参考系的分量。
所以我的问题实际上可以归结为：我必须使用什么矩阵才能在第 1 步执行其中的转换？
提前感谢您的时间和专业知识。
Real-life case (helps understand the question)
I am building a device that can freely rotate around all its axis (x, y, z) and is equipped with:
- an accelerometer (A) that gives me a vector pointing to the centre of the Earth (Va)
- a 3D magnetometer (M) that gives me the direction of the magnetic field of the Earth (Vm)
The two vectors share the same reference system (x, y, z), but what I am interested to find is the vector that points to the north relative to the Earth surface [think of a hand-held compass: I want to find where the needle should point to].
This video shows a "ball compass" that has pretty much the same behaviour that my device should replicate electronically.
The coding question
I did a bit of research, and it seems to me that I should use is 3D rotation matrices doing the following two steps:
- rotate the reference system of Vm from R to R', in such a way that y' will be parallel to Va,
- "flatten" the Vm vector setting its y' component to 0
Unluckily I am still confused on how I should proceed in concrete terms (I have no previous experience of working with vectors and matrices). One of the things that confuses me is that most of the material I could google, talks in terms of angles, but the data I am receiving from both sensors is in the form V(Vx, Vy, Vz), where Vz, Vy and Vz are the components of V along the reference system.
So my question really boils down to: what is the matrix I have to use in order to perform the transformation of which at step #1?
Thank you in advance for your time and expertise.
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我想到的一个更简单的答案是使用叉积。
你可以用 Ve = Va ⨉ Vm 找到东边
然后向北 Vn = Va ⨉ Ve
（这实际上可能是向南，没有考虑惯用手）
One simpler answer that comes to mind is to use cross products.
you can find east with Ve = Va ⨉ Vm
and then north with Vn = Va ⨉ Ve
(this might actually be south, haven't thought through the handedness)