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Hyperplane — Definition, Formula & Examples

A hyperplane is a flat subset of an nn-dimensional space that has dimension n−1n - 1. In R3\mathbb{R}^3 a hyperplane is an ordinary plane, and in R2\mathbb{R}^2 it is a line.

A hyperplane in Rn\mathbb{R}^n is the set H={x∈Rn∣aTx=b}H = \{\mathbf{x} \in \mathbb{R}^n \mid \mathbf{a}^T\mathbf{x} = b\}, where a∈Rn\mathbf{a} \in \mathbb{R}^n is a nonzero normal vector and b∈Rb \in \mathbb{R} is a scalar. This set forms an (n−1)(n-1)-dimensional affine subspace that divides Rn\mathbb{R}^n into two half-spaces.

Key Formula

aTx=a1x1+a2x2+⋯+anxn=b\mathbf{a}^T\mathbf{x} = a_1 x_1 + a_2 x_2 + \cdots + a_n x_n = b
Where:
  • a\mathbf{a} = Nonzero normal vector in \mathbb{R}^n, perpendicular to the hyperplane
  • x\mathbf{x} = Variable vector (x_1, x_2, \ldots, x_n) in \mathbb{R}^n
  • bb = Real scalar that determines the offset of the hyperplane from the origin

How It Works

A hyperplane is defined by a single linear equation in nn unknowns. The normal vector a\mathbf{a} is perpendicular to every direction that lies within the hyperplane. Changing bb shifts the hyperplane along the direction of a\mathbf{a} without tilting it. Every system of linear equations Ax=bA\mathbf{x} = \mathbf{b} can be viewed as the intersection of mm hyperplanes, one per row of AA. The solution set is whatever geometric object remains after intersecting all of them.

Worked Example

Problem: Describe the hyperplane in R3\mathbb{R}^3 defined by 2x1−3x2+x3=62x_1 - 3x_2 + x_3 = 6. Identify its normal vector and verify that the point (3,0,0)(3, 0, 0) lies on it.
Identify the normal vector: Read the coefficients of x1,x2,x3x_1, x_2, x_3 directly from the equation.
a=(2, −3, 1)\mathbf{a} = (2,\, -3,\, 1)
Check membership: Substitute (3,0,0)(3, 0, 0) into the left side of the equation.
2(3)+(−3)(0)+1(0)=62(3) + (-3)(0) + 1(0) = 6
Interpret geometrically: Since n=3n = 3, this hyperplane is a 2-dimensional plane. The vector (2,−3,1)(2, -3, 1) is perpendicular to that plane at every point on it.
Answer: The hyperplane is the plane 2x1−3x2+x3=62x_1 - 3x_2 + x_3 = 6 with normal vector a=(2,−3,1)\mathbf{a} = (2, -3, 1). The point (3,0,0)(3, 0, 0) satisfies the equation and therefore lies on the hyperplane.

Why It Matters

Solving a system Ax=bA\mathbf{x} = \mathbf{b} is geometrically the same as finding where mm hyperplanes intersect. Hyperplanes also appear in optimization (linear programming constraints) and machine learning (support vector machines find a separating hyperplane between data classes).

Common Mistakes

Mistake: Assuming a hyperplane always passes through the origin.
Correction: A hyperplane passes through the origin only when b=0b = 0. When b≠0b \neq 0 it is an affine subspace, not a linear subspace.