Abstract. In this study, we define the dual Fibonacci quaternion and the dual Lucas quternion. We derive the relations between the dual Fibonacci and the dual Lucas quaternion which connected the Fibonacci and the Lucas numbers. Furthermore, we give the Binet and Cassini formulas for these quaternions.

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In this paper, we introduce new adjoint curves which are associated curves in Euclidean space of three dimension. They are generated with the help of integral curves of special Smarandache curves. We attain some connections between Frenet apparatus of these new adjoint curves and main curve. We characterize these curves in which conditions they are general helix and slant helix. Finally, we exemplify them with figures

After its introduction by Grossman and Katz, the geometry part of geometric calculus has been mostly left unattended. In this paper, we give the elementary theorems to build the geometry on geometric space. We apply the well‐known theorems for inner product in geometric space. The terms geometric determinant and geometric vector product are introduced, and their properties are studied. We give the definitions of the circle, the plane, and the sphere and visualize them along with a line in geometric space. Finally, we discuss the Gram–Schmidt process in geometric space.