Discussion
The normals to surfaces represented by the Pfaffian differential equation \(P\,dx + Q\,dy + R\,dz = 0\) and the Lagrange's equation \(Pp + Qq = R\) :
Answer: B
The normals are orthogonal.
The surfaces given by \(P\,dx + Q\,dy + R\,dz = 0\) are those whose normal at each point is parallel to \((P, Q, R)\): every displacement \((dx, dy, dz)\) along such a surface is perpendicular to that vector.
The integral surfaces of Lagrange's equation \(Pp + Qq = R\) have normal \((p, q, -1)\), and the equation says \((P, Q, R) \cdot (p, q, -1) = 0\). So \((P, Q, R)\) is tangent to them and their normals are perpendicular to \((P, Q, R)\).
One family has normals along \((P, Q, R)\) and the other has normals perpendicular to it, so the normals of the two families are orthogonal, and the two families of surfaces cut each other at right angles. 'Collinear' and 'parallel' describe the same family, not the two.
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