Title: Comparison of Methods for Rotating a Point in $\mathbb{R}^3$: A Case Study Show Chinese title

Title: 比较旋转点的方法：$\mathbb{R}^3$案例研究

Abstract: This article presents and compares four approaches for computing the rotation of a point about an axis by an angle in $\mathbb{R}^3$. We illustrate these methods by computing, by hand, the rotation of point $P=(1,0,1)^T$ about axis $\mathbf{a}=(1,1,1)^T$ by angle $\theta=60^\circ$ (following the right-hand rule). The four methods considered are: (1) an ad hoc geometric method exploiting a symmetry in the situation; (2) a projection method that sets up a new coordinate system using the dot and cross products; (3) a matrix method which rotates the standard basis and uses matrix-vector multiplication; (4) a Geometric (Clifford) Algebra method that represents the rotation as a double reflection via a rotor. All methods yield the same exact result: $P'=\left(\tfrac{4}{3},\tfrac{1}{3},\tfrac{1}{3}\right)^T$. Show Chinese abstract

Abstract: 本文介绍了四种用于计算点绕轴旋转一定角度的方法，并对它们进行了比较。这些方法在三维空间中绕轴旋转的角度为$\mathbb{R}^3$时进行了说明。我们通过手动计算点$P=(1,0,1)^T$绕轴$\mathbf{a}=(1,1,1)^T$旋转角度$\theta=60^\circ$（遵循右手定则）来展示这些方法。 所考虑的四种方法分别是： (1) 利用情况中对称性的自定义几何方法； (2) 使用点积和叉积建立新坐标系的投影法； (3) 通过旋转标准基底并使用矩阵-向量乘法的矩阵法； (4) 使用几何（Clifford）代数表示旋转为两次反射的旋量法。 所有方法都得到了相同的确切结果： $P'=\left(\tfrac{4}{3},\tfrac{1}{3},\tfrac{1}{3}\right)^T$。