The set of hybrid numbers 𝕂 is a noncommutative number system that unified and generalized the complex, dual, and double (hyperbolic) numbers with the relation ih  =−hi=ε+i. Two hybrid numbers p and q are said to be similar if there exist a nonlightlike hybrid number x satisfying the equality x−1qx = p. And, it is denoted by p∼q. In this paper, we study the concept of similarity for hybrid numbers by solving the linear equations px = xq and qx−xp = c for boldp,boldq,boldc∈𝕂bold.

Quaternions are an important tool that provides a convenient and effective mathematical method for representing reflections and rotations in three-dimensional space. A unit timelike split quaternion represents a rotation in the Lorentzian space. In this paper, we give some geometric interpretations of split quaternions for lines and planes in the Minkowski 3-space with the help of mutual pseudo orthogonal planes. We classified mutual planes with respect to the casual character of the normals of the plane as follows; if the normal is timelike, then the mutual plane is isomorphic to the complex plane; if the normal is spacelike, then the plane is isomorphic to the hyperbolic number plane (Lorentzian plane); if the normal is lightlike, then the plane is isomorphic to the dual number plane (Galilean plane).

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The set of hybrid numbers is a noncommutative number system unified and generalized the complex, dual and double(hyperbolic) numbers with the relation ih = −hi = ε + i. Two hybrid numbers p and q are said to be similar if there exist a hybrid number x satisfying the equality x −1 qx = p. And it is denoted by p ∼ q. In this paper, we study the concept of similarity for hybrid numbers by solving the linear equations px = xq and qx − xp = c for p, q, c ∈ K.