Sliding Planes

Calculus Level pending

Two planes have the following equations (assuming \( p = (x, y, z) \) ):

\( n_1 \cdot (p - p_1(t) ) = 0 \)

with \(n_1 = (1, -1, 5) , p_1(t) = (3, 4, 3) + (0, 0, 1) t \) , and

\( n_2 \cdot (p - p_2(t) ) = 0 \)

with \( n_2 = (2, -1, 3), p_2(t) = (-1, 2, 5) + (1, 0, 0) t \)

\( t \) is the time parameter. The intersection of these two planes traces a plane. If the normal of the traced plane is along the vector \( (a, -b, c) \) where \( a, b, c \) are positive integers with \( gcd(a,b,c) = 1 \), find \( a + b + c \).

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