Reciprocal Basis Visualizer (2D Dual Basis & 3D Reciprocal Basis)

The "Finish Line" Analogy

Start with vector e₁. The condition e₁ · x = 1 defines a "Finish Line" — a hyperplane of all vectors satisfying this constraint.

Try This

Drag the Probe vector. It turns green when its tip touches the finish line!

The finish line is perpendicular to e₁, at distance 1/|e₁| from the origin. A longer e₁ brings the line closer.

The Second Master

Introduce e₂. The dual vector e¹ must now satisfy a second constraint:

e¹ · e₂ = 0

This forces e¹ to lie on the "Zero Line" — perpendicular to e₂.

The Constraint

e¹ must be orthogonal to every basis vector except e₁.

🎯 Challenge

Can you drag the Probe to where you think e¹ should be?
Hint: It must satisfy both constraints!

Check your answer in "The Conflict" tab!

The Visual Conflict

e¹ serves two masters simultaneously:

Job 1: Reach the Finish

e¹ · e₁ = 1

Job 2: Stay on Zero Line

e¹ · e₂ = 0

The unique solution is where these two lines intersect. This is the 2D version of:

e¹ = e₂⊥ / (e₁ · e₂⊥)

Vector Decomposition

Any vector v can be decomposed using both bases:

v = v¹e₁ + v²e₂ = v₁e¹ + v₂e²

Contravariant (vⁱ)

vⁱ = v · eⁱ — scalar, lives on eⁱ axis
vⁱ·e_i — component vector on e_i axis

Covariant (v_i)

v_i = v · e_i — scalar, lives on e_i axis
v_i·eⁱ — component vector on eⁱ axis

Show Decomposition

Contravariant: vⁱ and vⁱ·e_i

Covariant: v_i and v_i·eⁱ

The Basis Triad

Let {e₁, e₂, e₃} be a basis for V. These vectors span the space and form a parallelepiped.

Volume Condition

For a valid basis: e₁ · (e₂ × e₃) ≠ 0

The scalar triple product gives the signed volume of the parallelepiped. Adjust the vectors to see how volume changes.

The Reciprocal Basis

The reciprocal vectors {e¹, e², e³} are defined by:

e¹ = (e₂ × e₃) / V

e² = (e₃ × e₁) / V

e³ = (e₁ × e₂) / V

where V = e₁ · (e₂ × e₃)

Geometric Meaning

Each eⁱ is perpendicular to the other two basis vectors, scaled to satisfy eⁱ · e_i = 1.

Volume Duality

A beautiful relationship exists between the volumes of the two parallelepipeds:

e¹ · (e² × e³) = 1 / V

The Duality

V* = 1/V
The reciprocal volume is the inverse of the original!

Watch both parallelepipeds as you adjust the basis. When the original expands, its reciprocal contracts.

Biorthogonality

The defining property of reciprocal bases is the biorthogonality condition:

eⁱ · eⱼ = δⁱ_j

e₁ ·

1

0

e₂ ·

0

1

0

e₃ ·

0

1

This is the Kronecker delta: 1 when indices match, 0 otherwise. The matrix above shows live values!

Why It Works

Since e¹ = (e₂ × e₃)/V, it's perpendicular to both e₂ and e₃, hence e¹ · e₂ = e¹ · e₃ = 0.

Geometric Workspace

e₁

e₂

e¹

e²

v / Probe

Vector e₁

Angle 0°

Length 1.00

Vector e₂

Angle 90°

Length 1.00

Probe Vector

X 1.50

Y 0.50

Measurements

Probe · e₁ 0.00

e¹ · e₁ —

e¹ · e₂ —

|e¹| —

Area 1.00

Vector e₁

X 1.00

Y 0.00

Z 0.00

Vector e₂

X 0.00

Y 1.00

Z 0.00

Vector e₃

X 0.00

Y 0.00

Z 1.00

Display Options

Basis Vectors

Reciprocal Basis

Basis Parallelepiped

Reciprocal Parallelepiped

Drag vectors · Scroll to zoom