A Geometric Implication

Geometry Level 5

Let \(P_{1}\) be a plane determined by the vectors \(A\) and \(B\). Plane \(P_{2}\) be determined by vectors \(C\) and \(D\). Plane \(P_{3}\) be determined by vectors \(A\) and \(C\). Plane \(P_{4}\) be determined by vectors \(B\) and \(D\). If \(L_{1}\) represents the line of intersection (direction) of planes \(P_{1}\) and \(P_{2}\), and \(L_{2}\) represents the line of intersection(direction) of \(P_{3}\) and \(P_{4}\). The vector along the direction of \(L_{1} \times L_{2}\) is given by,

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