In this study, we show that the quaternion product of quaternionic operator whose scalar part is a real parameter and vector part is a curve in \({\mathbb {R}}^{3}\) and a spherical striction curve represents a slant ruled surface in \({\mathbb {R}}^{3}\) if the vector part of the quaternionic operator is perpendicular to the position vector of the spherical striction curve. In \({\mathbb {R}}^{3}\), exploitting this operator, we define the slant ruled surface corresponding to the natural lift curve on the subset of the tangent bundle of unit 2-sphere, \(T{\bar{M}}.\) Then, we classify \(\vec {{\bar{q}}}-, \vec {{\bar{h}}}-\) and \(\vec {{\bar{a}}}-\) slant ruled surfaces. Furthermore, these surfaces can also be expressed with 2- parameter homothetic motions. Finally, we give the geometric interpretations of this operator with some examples.

在这项研究中，我们证明了四元数算子的四元数乘积，其标量部分是实参数，向量部分是\({\mathbb {R}}^{3}\)中的曲线，球面收缩曲线表示倾斜\({\mathbb {R}}^{3}\)中的直纹曲面，如果四元数算子的向量部分垂直于球面收缩曲线的位置向量。在\({\mathbb {R}}^{3}\)中，利用该算子，我们定义了与单位 2 球体切丛子集上的自然升力曲线对应的斜直纹面，\(T{ \bar{M}}.\)然后，我们对\(\vec {{\bar{q}}}-, \vec {{\bar{h}}}-\)和\(\vec {{\酒吧{a}}}-\）倾斜直纹表面。此外，这些表面也可以用 2 参数相似运动来表示。最后，我们通过一些例子给出了这个算子的几何解释。