Find the vector lines and the family of surfaces normal to…
Find the vector lines and the family of surfaces normal to the vector lines of \[ \vec A = x\,\vec i + y\,\vec j - z\,\vec k. \]
Explanation of Method
The characteristic system separates immediately into two single-variable-ratio ODEs, giving two first integrals directly; the vector lines are then their intersection curves, and any function of the two first integrals defines a normal surface.
Step 1 · Set up the characteristic system
\[ \frac{dx}{x}=\frac{dy}{y}=\frac{dz}{-z}. \]
Step 2 · First integral
From the first two ratios, $d(\ln(y/x))=0 \implies u_1=y/x=C_1$.
Step 3 · Second integral
From the first and third ratios, $d(\ln xz)=0 \implies u_2=xz=C_2$.
Step 4 · Describe the vector lines
Each vector line is the intersection of the plane $y=C_1x$ with the hyperbolic cylinder $xz=C_2$: \[ y=C_1x,\qquad z=\frac{C_2}{x}. \] Equivalently, $x=x_0e^t$, $y=y_0e^t$, $z=z_0e^{-t}$, which manifestly keeps $y/x$ and $xz$ constant.
Verification
\[ \vec A\cdot\nabla u_1 = x\!\left(-\frac{y}{x^2}\right)+y\cdot\frac1x = 0. \checkmark \] \[ \vec A\cdot\nabla u_2 = xz+(-z)x = 0. \checkmark \] For any $F(u_1,u_2)$, $\vec A\cdot\nabla F=F_{u_1}(\vec A\cdot\nabla u_1)+F_{u_2}(\vec A\cdot\nabla u_2)=0$, confirming every such surface is normal to $\vec A$.
Result
The vector lines and the family of normal surfaces are: