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Normal Vector — Definition, Formula & Examples

A normal vector is a vector that points perpendicular (at a right angle) to a surface or curve at a specific point. It tells you which direction is 'straight out' from the surface at that location.

Given a surface defined by F(x,y,z)=0F(x, y, z) = 0, the normal vector at a point on the surface is n=∇F=⟨∂F∂x, ∂F∂y, ∂F∂z⟩\mathbf{n} = \nabla F = \left\langle \frac{\partial F}{\partial x},\, \frac{\partial F}{\partial y},\, \frac{\partial F}{\partial z} \right\rangle, which is orthogonal to every tangent vector at that point. For a plane with equation ax+by+cz=dax + by + cz = d, the normal vector is n=⟨a,b,c⟩\mathbf{n} = \langle a, b, c \rangle.

Key Formula

n=∇F=⟨∂F∂x, ∂F∂y, ∂F∂z⟩\mathbf{n} = \nabla F = \left\langle \frac{\partial F}{\partial x},\, \frac{\partial F}{\partial y},\, \frac{\partial F}{\partial z} \right\rangle
Where:
  • n\mathbf{n} = The normal vector to the surface
  • F(x,y,z)F(x,y,z) = A scalar function whose level surface F = 0 defines the surface
  • ∇F\nabla F = The gradient of F, computed via partial derivatives

How It Works

For a plane, you can read the normal vector directly from the coefficients of xx, yy, and zz in the plane equation. For a general surface F(x,y,z)=0F(x,y,z) = 0, compute the gradient ∇F\nabla F and evaluate it at the point of interest. If you have two tangent vectors lying in a plane or on a surface, their cross product gives a normal vector. The normal vector is not unique in direction — you can scale it by any nonzero scalar, and both n\mathbf{n} and −n-\mathbf{n} are valid normals pointing in opposite directions.

Worked Example

Problem: Find a normal vector to the plane 3x − 2y + 5z = 10.
Identify coefficients: For a plane ax + by + cz = d, the normal vector is simply the vector of coefficients.
n=⟨a,b,c⟩\mathbf{n} = \langle a, b, c \rangle
Read off the normal: From the equation 3x − 2y + 5z = 10, extract a = 3, b = −2, c = 5.
n=⟨3,−2,5⟩\mathbf{n} = \langle 3, -2, 5 \rangle
Answer: A normal vector to the plane is n=⟨3,−2,5⟩\mathbf{n} = \langle 3, -2, 5 \rangle.

Why It Matters

Normal vectors are essential in multivariable calculus for computing surface integrals and flux. In computer graphics, normal vectors at each point on a surface determine how light reflects, controlling shading and rendering. Physics uses them to define forces on surfaces, such as the normal force in mechanics.

Common Mistakes

Mistake: Confusing a normal vector with a normalized (unit) vector.
Correction: A normal vector is perpendicular to a surface and can have any magnitude. A normalized vector has magnitude 1. To get a unit normal, divide the normal vector by its magnitude: n^=n∥n∥\hat{\mathbf{n}} = \frac{\mathbf{n}}{\|\mathbf{n}\|}.