Dated: 30-10-2024
Orthogonal Surfaces
Normal line
If \(P_0(x_, y_0, z_0)\) be any point on the surface \(f(x, y, z) = 0\).
If \(f(x, y, z)\) is differentiable at \(P_0\) then the normal line at \(P_0\) has the equation
\[x = x_0 + f_x(P_0)t\]
\[y = y_0 + f_y(P_0)t\]
\[z = z_0 + f_z(P_0)t\]
Example
Find the tangent plane1 and normal line to the surface
\[f(x,y,z)=x^{2}+y^{2}+z^{2}-14\]
at
\[P(1,-2,3)\]
Solution
\[f_x = 2x, \, f_x(P_0) = 2(1) = 2\]
\[f_y = 2y, \, f_y(P_0) = 2(-2) = -4\]
\[f_z = 2z, \, f_z(P_0) = 2(3) = 6\]
Tangent Plane1
\[2(x-1)-4(y+2)+6(z-3)=0\]
\[x-2y+3z-14=0\]
Normal line
\[
\frac{x-1}{2}=\frac{y+2}{-4}=\frac{z-3}{6}
\]
Multiplying by \(2\).
\[
\frac{x-1}{1}=\frac{y+2}{-2}=\frac{z-3}{3}
\]
Orthogonal Surfaces
Two surfaces are called orthogonal if, at their point of intersection, normals of both are orthogonal.
Condition
Imagine we have 2 surfaces
\[f(x, y, z) = 0\]
\[g(x, y, z) = 0\]
Their normals are
\[\vec{n_1} = f_x \hat i + f_y \hat j + f_z \hat k\]
\[\vec{n_2} = g_x \hat i + g_y \hat j + g_z \hat k\]
Using the dot product2 definition
\[\vec{n_1} \cdot \vec{n_2} = 0\]
\[f_{x}g_{x}+f_{y}g_{y}+f_{z}g_{z}=0\]
References
Read more about notations and symbols.
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Read more about tangent lines. ↩↩