# 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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