Vector Space Explorer~Finite Vectors

About the simulation

Physics / math

This simulation explores finite-dimensional vector-space operations in 2D (\(\mathbb{R}^2\)): linear combination \(\mathbf{v}=c_1\mathbf{a}+c_2\mathbf{b}\), dot product \(\langle\mathbf{a},\mathbf{b}\rangle\), vector angle, and projection \(\mathrm{proj}_{\mathbf{b}}(\mathbf{a})\).

Simulation visuals

The canvas shows vectors a, b, and the result vector, with optional overlays for basis vectors \(\hat e_1,\hat e_2\), parallelogram construction, projection guide, and angle arc. Coordinates and computed results update live.

Controls

Drag vector tips on the canvas (snap: 0.5), or edit components/coefficient inputs in the sidebar. Use mode tabs to switch between Linear Combination and Dot Product & Projection, toggle overlays via checkboxes, scroll to zoom, and click Reset All to restore defaults.

Dark

🌙

±10.0

Drag the colored tips of vectors a and b on the canvas. Use scroll wheel to zoom in/out. Values snap to 0.5 when dragging.

Guided Inquiry

1 · Vectors and their components

Let’s start with the big idea. Two vectors a and b in the plane generate the combinations v = c₁a + c₂b. The pair (c₁, c₂) isn’t the vector — it’s the vector’s representation in the set {a, b}. A quantum state works the same way: |ψ⟩ is one object, and its components depend on the basis you expand it in.

2 · What two vectors reach

Let’s see how far two vectors can reach. Here a = (3, 1) and b = (1, 3). Now suppose b were changed to (6, 2) — exactly twice a. Which vectors do you think could then be written as c₁a + c₂b?

3 · Same coefficients, different basis

Time to swap the building blocks. With a = (3, 1), b = (1, 3) and c₁ = c₂ = 1, the readout gives v = (4.0, 4.0). Now let a = ê₁ = (1, 0) and b = ê₂ = (0, 1), keeping c₁ = c₂ = 1. What do you think v becomes?

4 · Projection and the length of b

Now let’s turn to projections. With a = (3, 1) and b = (1, 3), the projection of a onto b is proj = (0.60, 1.80). If you replace b by 2b = (2, 6), what do you expect the projection to be?

5 · When a coefficient is an inner product

One last question. The projection is proj = (⟨a, b⟩ / ⟨b, b⟩) b. Suppose b is replaced by a vector of unit length, ‖b‖ = 1. What do you think the number multiplying b becomes?

Listen

Vectors

Max coordinate value: ±10

a = ( , )

b = ( , )

Linear combination

v = a + b

c₁ c₂

Dot product

Angle between vectors

Projection of a onto b

Display options

Standard basis ê₁, ê₂ Show parallelogram Show projection Show angle arc

The Physics

Finite vector spaces · what this sim shows, the physics behind it, and every control

At a glance

Two arrows a and b in a plane can be mixed with a pair of numbers to build new vectors, and those numbers change when the arrows do. This sim lets you build vectors, test when two arrows are enough, and measure how much of one lies along another: the real-number groundwork for expanding a quantum state in a basis.

What you'll learn

How a vector is built from two others, and what the coefficients c₁ and c₂ tell you.
When two vectors are enough to reach the whole plane, and how the determinant tests it.
How the inner product gives lengths and angles, and how a projection splits a vector into two parts.
Which kind of basis lets you read a component straight off with an inner product.

The Linear Combination view: v is the diagonal of the parallelogram spanned by c₁a and c₂b.

What's on screen

a and b (blue, green): the two vectors. Drag their pulsing tips to move them.
v (pink): the combination c₁a + c₂b, with its dashed parallelogram (Linear Combination view).
ê₁ and ê₂ (sky, violet): the standard basis, (1, 0) and (0, 1).
proj (amber) and a violet angle arc: the projection of a onto b and the angle θ (Dot Product & Projection view).
Sidebar readouts: the arithmetic behind each view, line by line.

See itOpen the sim at its starting pair, a = (3, 1) and b = (1, 3), with c₁ = c₂ = 1. Watch for: the pink v closing the dashed parallelogram, and the readout showing v = (4.0, 4.0).

The controls

Press ⌖ on a row to see where that control sits in the sim.

Control	What it changes	Range
Vectors a, b	The components of a and b. Type them, or drag the tips on the canvas (dragging snaps to 0.5).	−10 to 10 opens at a = (3, 1), b = (1, 3)
View tabs	Switch between Linear Combination and Dot Product & Projection. Each has its own readouts and overlays.	2 views opens on Linear Combination
c₁, c₂	The coefficients in v = c₁a + c₂b. Linear Combination view only.	any number opens at 1 and 1
Standard basis ê₁, ê₂	Draws the two unit arrows (1, 0) and (0, 1) at the origin, in both views.	on / off opens on
Show parallelogram	Draws the parallelogram whose diagonal is v. Linear Combination view only.	on / off opens on
Show projection	Draws proj, the part of a along b, and the dashed guide from a to it. Dot Product & Projection view only.	on / off opens on
Show angle arc	Draws the arc for the angle θ between a and b. Dot Product & Projection view only.	on / off opens on
Readouts	The sidebar arithmetic. Linear Combination view: v and det(a, b), with a basis check. Dot Product & Projection view: ⟨a, b⟩, θ and the projection.	panel
Zoom	Scroll over the canvas to zoom. Changing a vector re-fits the view.	about ±1 to ±55
Reset	Restores the opening vectors, c₁ = c₂ = 1 and all four overlays, and returns to Linear Combination.	top bar

Fixed: the space is real (ℝ²), so every component and coefficient is a real number. Nothing animates, so there is no Play button.

Components belong to a basis

A linear combination takes c₁ of a and c₂ of b and adds them tip to tail. The sim adds one coordinate at a time:

The sim's Linear Combination readout.

The pair (c₁, c₂) is a recipe written for the set {a, b}. It is not the vector. With a = (3, 1), b = (1, 3) and c₁ = c₂ = 1 the recipe builds v = (4, 4).
The same recipe with {ê₁, ê₂} builds (1, 1), a different arrow. Going the other way, the arrow (4, 4) needs c₁ = c₂ = 1 in {a, b} but c₁ = c₂ = 4 in {ê₁, ê₂}.

Key idea. The arrow is the vector. Its components are its coefficients in one chosen basis, and they change when the basis changes.

See itBuild v from a and b with c₁ = c₂ = 1, then swap a and b for ê₁ and ê₂ and keep the same c₁ and c₂. Watch for: the readout changing from v = (4.0, 4.0) to v = (1.0, 1.0).

Two vectors reach the plane only if independent

The set of everything c₁a + c₂b can reach is the span of {a, b}. If b = 2a, then c₁a + c₂b = (c₁ + 2c₂)a, so every combination lies on one line.

The test the sim prints under the combination.

det(a, b) = aₓb_y − a_ybₓ. Its size |det| is the area of the parallelogram, and det = 0 means that area has collapsed.
When det ≠ 0, {a, b} is a basis: every vector in the plane has exactly one pair (c₁, c₂). At the opening pair, det = 3·3 − 1·1 = 8.
The sim shows ✓ valid basis, and switches to ✕ linearly dependent once |det| falls below 0.05.

See itSet b = 2a = (6, 2) with a = (3, 1). Watch for: det reading 0.00, the verdict flipping to ✕, and the warning "Vectors are collinear" on the canvas.

The inner product gives length and angle

Inner product, angle and length. θ runs from 0° to 180°.

⟨a, b⟩ is one number built from two vectors. Its sign says whether they lean the same way (positive) or opposite ways (negative).
Dividing by the lengths gives cos θ = ⟨a, b⟩ / (‖a‖‖b‖), the angle the sim reports.
For two non-zero vectors, ⟨a, b⟩ = 0 means they are orthogonal, θ = 90°.

a_x a_y b_x b_y

These sliders move the sim too.

See itSet a = (3, 1) and b = (−1, 3). Watch for: ⟨a, b⟩ reading 0.00 and θ reading 90.00°.

Projection takes the part of a along b

The projection, and the leftover that is perpendicular to b.

The sim's opening pair: ⟨a, b⟩ = 6, ⟨b, b⟩ = 10, scalar 0.6, proj = (0.60, 1.80).

proj is the point on the line through b nearest to a. What is left, a − proj = (2.40, −0.80), is perpendicular to b. That is the dashed guide.
Dividing by ⟨b, b⟩ = ‖b‖² removes the length of b. Only its direction matters: replace b by 2b and the scalar halves, so proj stays put.
If b = (0, 0) there is no direction to project onto. The sim prints proj = (0.00, 0.00) and θ = 0.00° as placeholders.

See itProject a = (3, 1) onto b = (1, 3). Watch for: the dashed guide meeting b at a right angle, and proj = (0.60, 1.80).

See itReplace b = (1, 3) by 2b = (2, 6). Watch for: the scalar halving from 0.600 to 0.300 while proj stays (0.60, 1.80).

In an orthonormal basis components are inner products

An orthonormal basis: perpendicular, each of length 1.

If ‖b‖ = 1, then ⟨b, b⟩ = 1 and the projection scalar is just ⟨a, b⟩. Perpendicular unit vectors make every component an inner product.
ê₁ and ê₂ are orthonormal, so the standard components of v are ⟨ê₁, v⟩ and ⟨ê₂, v⟩.
The opening pair {a, b} is not orthonormal. For v = (4, 4), ⟨a, v⟩ = 16, yet c₁ = 1.
Quantum mechanics uses the same step: |ψ⟩ = Σ cₙ|n⟩ with cₙ = ⟨n|ψ⟩ for an orthonormal basis {|n⟩}. This sim is real and shows no probabilities; the complex case adds a conjugate in the inner product.

See itKeep a = (3, 1) and make b the unit vector ê₁ = (1, 0). Watch for: the scalar reading 3.000, the same number as ⟨a, b⟩ = 3.00.

Where it matters

Polarizers. A polarizer keeps the component of the light's polarization along its axis, and the transmitted intensity is I₀cos²θ (Malus's law). Take a as the polarization and b as the axis: the opening pair has θ = 53.13°, which passes 0.36 of I₀.
Spin-½ and qubits. The state lives in a complex two-dimensional space. It is written in a chosen basis, such as spin up and spin down, and the squared moduli of its two components are the outcome probabilities.
Expanding in eigenstates. A state is expanded in the orthonormal eigenstates of the quantity you measure, with coefficients cₙ = ⟨n|ψ⟩.

The sim itself models none of these. It shows the real-vector geometry they share.

Go deeper: Malus's law from the inner product

The numbers are the opening pair's: ⟨a, b⟩ = 6 and ⟨a, a⟩ = ⟨b, b⟩ = 10.

The same expansion in Dirac notation, for a normalised state.

Summary

In one line. A vector is an arrow, and its components are the coefficients of one basis; inner products give lengths, angles and projections, and an orthonormal basis turns components into inner products.

Key points

{a, b} is a basis of the plane exactly when det(a, b) ≠ 0.
⟨a, b⟩ = ‖a‖‖b‖cos θ, and ⟨a, b⟩ = 0 means perpendicular.
proj = (⟨a, b⟩ / ⟨b, b⟩) b, and a − proj is perpendicular to b.

Common mix-ups

"The pair (c₁, c₂) is the vector." It is the vector's components in one basis. The same pair in ê₁, ê₂ builds (1, 1), not (4, 4).
"Any two arrows reach the whole plane." Only if det ≠ 0. At b = 2a every combination lies on one line.
"A longer b captures more of a." The projection depends on the direction of b only. Doubling b halves the scalar.

Things to try on your own

With a = (3, 1) and b = (1, 3), find c₁ and c₂ that make v = (4, 0). One of them is negative.
Set a = (2, 0) and b = (0, 3), then move b's tip until ⟨a, b⟩ turns negative. Note the sign of the scalar.
Compare proj for b = (1, 3) and b = (−1, −3), and explain the sign of the scalar.