Tangent and Normal
For any curve \(\vec{r}(t)\), the velocity vector \(\vec{v}(t)\) is tangent to the path. A vector that is perpendicular to the tangent is called a Normal Vector \(\vec{N}\).
For a surface \(f(x, y, z) = c\), the gradient vector \(\nabla f\) is always normal (perpendicular) to the surface.
Worked Examples
Example 1: Normal to a Surface
Find the normal vector to the sphere \(x^2 + y^2 + z^2 = 9\) at the point \((1, 2, 2)\).
- Let \(f(x, y, z) = x^2 + y^2 + z^2\).
- \(\nabla f = \langle 2x, 2y, 2z \rangle\).
- At \((1, 2, 2)\), \(\nabla f = \langle 2, 4, 4 \rangle\).
- Result: \(\langle 2, 4, 4 \rangle\). Any vector pointing straight out from the center is normal to the sphere.
The Bridge to Quantum Mechanics
Normal vectors are used to define Boundary Conditions. When a particle hits a wall, we often need to calculate the component of its momentum normal to the wall. In Quantum Mechanics, the probability current \(\vec{J}\) must be continuous. If a particle is confined in a box, the component of \(\vec{J}\) normal to the wall must be zero (\(\vec{J} \cdot \vec{n} = 0\)). This "normal" logic is what forces the wavefunction to drop to zero at the edges, creating the discrete energy levels of a trapped particle.