We propose two new approaches for the tangential intersection of two surfaces in Euclidean 3-space. Our new methods which differs from the existing methods based on a new operator and the Rodrigues' rotation formula. Our first method is also depending on some new defined derivative operators. The second one which is given without any derivative operator can be easily applied to all kinds of tangential intersection problems. By using the Rodrigues' formula and the defined new operators, we obtain the curvature and the tangent, principal normal, and binormal vectors of the tangential intersection curve. We show that our new operator and Rodrigues' rotation formula can also be used for all kinds of transversal intersections. (C) 2016 Elsevier B.V. All rights reserved.