# Constructing Normals to Surfaces

Recall that a surface is defined by a set of vectors depending on two parameters. For example, r(u, v) = < 1.1 cos v sin u, 1.1 sin v sin u, 1.1 cos u> defines the surface of a sphere of radius 1.1 centered at the origin. Because r_u and r_v are vectors in the direction of tangent lines to the surface, the vector r_u x r_v is in the direction normal to the surface. Here is how one plots a normal line to this surface when u = pi/4 and v = 0.

<<Calculus`VectorAnalysis`

r = {1.1 Cos[v] Sin[u], 1.1 Sin[v] Sin[u], 1.1 Cos[u]};

ru = D[r, u];

rv = D[r, v];

normal = CrossProduct[ru, rv];

u0 = Pi/4; v0 = 0;

point = r /. {u -> u0, v -> v0};

spnorm = normal /. {u -> u0, v -> v0};

graph1= ParametricPlot3D[Evaluate[r], {u, 0, Pi/2}, {v, 0, 2Pi}];

graph2=Graphics3D[Line[{point, point + 1.5 spnorm}]];

output = Show[graph1, graph2, AspectRatio->Automatic]