Inner Product Invariance Under Coordinate Rotation

Source

Lecture: Lecture 2 — Vectors (Euclidean space, metric, inner product, the velocity vector).
Simulation: Simulation 1 — lecture visual aid: vectors A and B shown in two panels whose axes differ by a rotation θ; each panel shows the components and the inner product A·B; a slider sets θ; tips are draggable.
Concept: The inner product of two vectors — and therefore each vector's norm — is invariant under rotation of the coordinate axes, even though the individual components change.
Serves: Euclidean space as a geometric structure equipped with a metric; the Kronecker delta δ_ij as the Cartesian metric; computing inner products and norms; the velocity vector as a coordinate-independent geometric object.
Inquiry: Rotate the frame — which of these changes: the components A_x, A_y; the length |A|; or the inner product A·B? Commit a prediction before running.
Misconception: That rotating the coordinate axes changes the inner product or a vector's length (confusing component change with geometric change).

Pedagogy principles applied

ID	Principle	How it shows up here
A1	One concept	Only inner-product invariance under rotation; the velocity question is a side-panel prompt, not a second sim.
A3/A4	Formalism staged & toggleable	Full A·B expansion is in the core readout but revealed at Stage 3; the δ_ijAⁱB^j notation lives in The Physics revision sheet, opened from the top bar.
B3	Define every symbol	Notation panel defines A_x, A_x′, θ, δ_ij, \|A\| on first use.
B4	Descriptive headers	"Frame S (fixed axes)" and "Frame S′ (axes rotated by θ)".
C1	Every number referenced	Components compared across panels; the two A·B values compared for equality; norms read as invariant.
C2	Numbers carry context	"components in Frame S / S′", primed vs unprimed subscripts, dimensionless grid units.
D1	Visualization dominant	The two-panel canvas is the largest element.
D2	Readable text	Canvas labels ≥13px, readouts ≥12.5px, side prose ≥14px.
D3	Labelled arrows	Thick coloured arrows with adjacent labels A and B, not bare dots.
D5	Graceful extremes	θ=0° → panels identical; zero/negative components handled; tips clamped inside bounds.
D7	Additive disclosure	Stages only reveal readout rows; slider & dragging work from load, nothing is disabled.
D8	Sentences off-canvas	Canvas carries labels/numbers only; explanations live in side panels.
G2	Invariants flagged	A·B boxes are colour-matched; norms tagged "invariant".
G3	Real re-render	θ rotates the S′ grid, axes, projections; component numbers update — not a label swap.
G4c	Identity preserved	A and B keep identity, colour and geometric direction across both panels.
H1	Misconception structural	"\|A\| changes" / "A·B changes" are prediction choices, not a standing warning.
H2	Targeted rebuttal	"The arrow hasn't moved — only the grid rotated" appears only on selecting the wrong choice.
J1	Consistent notation	Roman subscripts, primed frame x′ y′, angle θ throughout.
J3	Correct physics	Passive rotation matrix; A·B and norms exactly equal to displayed precision in both frames.

Inner Product Invariance

Teach me

Speed

Click and drag points to move.

Frame S (fixed axes)

P (–, –) point P in Frame S

tail (–, –)

tip (–, –)

V V_x=– V_y=–

u u_x=– u_y=–

‖V‖ = – ‖V‖ = √(V_x²+V_y²) = – = –

‖u‖ = – ‖u‖ = √(u_x²+u_y²) = – = –

u·V = – u·V = u_xV_x+u_yV_y = – = –

Frame S′ (axes rotated by θ = 30°)

P (–, –) point P in Frame S′

tail (–, –)

tip (–, –)

V V_x′=– V_y′=–

u u_x′=– u_y′=–

‖V‖ = – ‖V‖ = √(V_x′²+V_y′²) = – = –

‖u‖ = – ‖u‖ = √(u_x′²+u_y′²) = – = –

u·V = – u·V = u_x′V_x′+u_y′V_y′ = – = –

Guided Inquiry

1 · Two frames, one plane

Let’s get our bearings. Both panels show the same plane; the S′ axes are simply turned counter-clockwise from the S axes by an angle θ. Nothing drawn in the plane is ever moved between the panels — only the axes it’s measured against. Each panel reports the coordinates of the point P, and later on the components, the lengths and the inner product u·V of two arrows.

2 · Predict — the length of V

Here’s your first prediction. The arrow V is drawn just once, and each panel reads its two components against its own axes — so those two pairs of numbers differ. What do you think happens to ‖V‖?

3 · Predict — the inner product

Now a second arrow, u, joins the first. Each panel forms u·V = u_xV_x + u_yV_y from its own components, and every one of those components differs between the panels. What do you think — the two results are…

4 · Predict — when u·V is zero

Try a special case. Suppose u is dragged until the boxed u·V reads 0.00 in Frame S. What do you think Frame S′ would show then?

Now go ahead and do it: drag the tip of u until u·V reads 0.00, and compare the two panels.

5 · Invariants

So let’s give this a name. A number that every rotated frame agrees on is called an invariant: ‖V‖, ‖u‖ and u·V are invariants, while the separate components aren’t. If V is a velocity, the components depend on how you lay down your axes, but ‖V‖ — the speed — doesn’t. Press ▶ Play to sweep θ and compare the two kinds of row.

Listen

Controls

▾

Rotation angle θ θ = 30°

0°360°

Include calculation Show projection

Disclosure stage

Invariants

▾

‖V‖ = –

‖u‖ = –

u·V = –

Notation

▾

P — a point; its coordinates (x, y) in S differ from (x′, y′) in S′ when θ ≠ 0 or the S′ origin is translated.
V_x — x-component of V in Frame S; V_x′ — x′-component in Frame S′. Components depend on the frame; ‖V‖ does not.
θ — rotation angle of the S′ axes relative to S (counter-clockwise positive). t = (t_x, t_y) — translation of the S′ origin in S.
— Kronecker delta (Cartesian metric). ‖V‖ = ; u·V is invariant under rotation and translation.

Formal tools

Inner product via the Cartesian metric

Kronecker delta — the metric δ_ij

Passive rotation of components (axes turn by +θ)

Norm (length) from the inner product

Why it is invariant — the cross terms cancel exactly

The inner product is the contraction of the components with the metric δ_ij. In Cartesian coordinates δ_ij is the same in every rotated frame, so although the components transform, the scalar δ_ijuⁱV^j does not — that is precisely the equality of the two boxed values.

The Physics

Inner product invariance · what this sim shows, the physics behind it, and every control

At a glance

Two arrows lie in one plane, and two frames, S and S′, measure them against axes that differ by a rotation θ. The sim shows what each frame reads for the components, the lengths and the product u·V, so you can sort out which numbers describe the arrows and which describe the axes.

What you'll learn

How the components of a vector in a frame depend on the rotation angle θ of the axes.
How the length ‖V‖ and the inner product u·V are computed in each frame, and how they depend on θ.
How the angle between two arrows is related to u·V.
What shifting the origin of the axes does to point coordinates and to vector components.

The same arrows drawn in both panels. Only the axes differ.

What's on screen

Frame S (left): fixed grey axes x, y.
Frame S′ (right): violet axes x′, y′ turned by θ, with the arc marking θ.
Arrow V (blue) and arrow u (amber, stage 4): drawn with dashed projection lines onto each frame's own axes.
Readouts under each panel: components, ‖V‖, ‖u‖ and u·V. The sidebar's Invariants panel lists the same three at stage 4.

See itOpen both panels at θ = 30°. Watch for: the same two arrows in both panels, with the S′ axes turned 30° from the S axes.

The controls

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

Control	What it changes	Range
Rotation angle θ	Turns the S′ axes counter-clockwise from the S axes. ▶ Play sweeps it by itself, at about 14° per second at 1×.	0°–360° opens at 30°
S′ origin offset t_x, t_y	Slides the origin of the S′ axes away from the S origin, without turning them. You can also drag the S′ origin dot.	−4 to 4 opens at 0, 0
Disclosure stage	1: a single point P. 2: arrow V from the origin. 3: V with its tail moved off the origin. 4: adds the second arrow u and the Invariants panel.	1–4 opens at 1; Free exploration starts at 4
Include calculation	Writes out each panel's own arithmetic next to its ‖V‖, ‖u‖ and u·V.	on / off opens off
Show projection	Draws dashed lines from each arrow's tip onto the frame's own axes. Turning it off also hides the tail, tip and component rows.	on / off opens on
Dragging	Drag P (stage 1), the tip of V (stages 2–4), the tail of V (stage 3) or the tip of u (stage 4), in either panel. Dragging the S′ origin dot sets t. Positions stop at ±5.5.	canvas
Readouts	Under each panel: P, tail and tip, components, ‖V‖, ‖u‖ and u·V, as many rows as the stage allows.	two panels
Invariants	The sidebar panel that lists ‖V‖, ‖u‖ and u·V, computed in frame S.	stage 4
▶ Play / ⏸ Pause	Turns θ continuously, or holds it where it is.	top bar
Speed	How fast θ turns while playing.	0.25×–4×
↻ Reset	Restores θ, the offset, the arrows and both checkboxes to their opening values, and pauses.	top bar

Fixed: lengths are in grid units. The opening arrows are V = (3, 4) and u = (4, −1) in S. At stage 3, V's tail starts at (−1.5, 1) and its tip at (3, 4), so V = (4.5, 3) there.

What turning the axes changes

Both panels show the same plane. S′ is S with its axes turned counter-clockwise by θ. Nothing in the plane moves between the panels; only the axes it is measured against do.

A frame reads a component by dropping a perpendicular from the arrow's tip onto its own axis. These are the dashed projection lines.
Turn the axes and the feet of those lines slide along them. The numbers change, and the arrow does not.

The sim's rule for S′ components from S components, for any arrow A (V or u). The axes turn; the arrow stays put.

With the opening V = (3.00, 4.00), θ = 30° gives (4.60, 1.96) in S′. At θ = 90° it reads (4.00, −3.00): x′ = V_y and y′ = −V_x. At θ = 0° or 360° the two panels agree.

See itCompare V's components in the two panels at θ = 30°. Watch for: V_x, V_y and V_x′, V_y′ are different numbers, while the arrow looks the same in both panels.

Length and inner product agree in both frames

Each frame forms the inner product and the length from its own components, using the Cartesian metric δ_ij (1 when i = j, 0 otherwise):

The lecture figures write |V| for the length ‖V‖ that the sim prints.

The components differ between the panels, yet ‖V‖, ‖u‖ and u·V come out equal in S and S′. This holds at every θ, not only at multiples of 90°.

The sim at its opening settings, θ = 30°, with V = (3, 4) and u = (4, −1).

θ t_x t_y

These sliders move the sim too.

See itLet θ sweep all the way round. Watch for: the components keep changing, while the u·V values for S and S′ stay equal to each other, and so do the two ‖V‖ values.

Why the inner product survives the turn

Multiply out the rotated components of any two arrows A and B. The cross terms cancel, and cos²θ + sin²θ = 1 collects the rest.

A rotation changes the components, but it leaves δ_ij unchanged. A number built by contracting components with δ_ij therefore cannot depend on θ.
Geometrically, u·V = ‖u‖ ‖V‖ cos φ, where φ is the angle between the two arrows. No turn of the axes opens or closes that angle. With the opening arrows, 5.00 × 4.12 × cos 67.2° = 8.00.
So for two non-zero arrows, u·V = 0 exactly when they are perpendicular, and both frames agree on it.

See itDrag the tip of u until it is perpendicular to V. Watch for: both u·V boxes read 0.00 while the arrows meet at a right angle in both panels.

Moving the origin changes points, not vectors

The offset t moves the S′ origin. The S′ coordinates of every point (P, and the tail and tip of V) change, because they are measured from the new origin.
A vector is the difference tip − tail, so t cancels. Its components, ‖V‖ and u·V stay as they were.

At stage 3 with θ = 30°, V = tip − tail = (4.5, 3.0) in S and (5.40, 0.35) in S′, whatever t is.

See itSlide the S′ origin from (0, 0) to (2, −1.5) at stage 3. Watch for: the tip's S′ coordinates change, while V_x′, V_y′ and ‖V‖ do not.

Where it matters

Lecture 2(d). Alice's axes are Bob's turned by 45°, so her x′ axis lies along the velocity. The lecture notes mark her basis with tildes; this figure and the sim use primes.

Speed. Bob and Alice track one particle with axes 45° apart. Their velocity components differ, but ‖V‖² = v² for both, so they agree on the speed.
Distance. The separation of P = (0, 0) and Q = (1, 1) in Bob's axes becomes (√2, 0) in Alice's. Both get √2.
The next step in the course. Keep the recipe but change the metric, with a relative minus sign between the time and space terms. The number that stays fixed is then the spacetime interval.

Go deeper: the matrix form of the cancellation

Write the S′ components as R times the S components, with R = (cos θ, sin θ; −sin θ, cos θ). Then u′·V′ = (Ru)ᵀ(RV) = uᵀ(RᵀR)V.
Because RᵀR = 1 for a rotation, u′·V′ = uᵀV. That is the cancellation of the previous section in one line.

Summary

In one line. Turning the axes changes an arrow's components. The length ‖V‖ and the inner product u·V, built with the Cartesian metric δ_ij, take the same value in every frame.

Key points

S′ components come from S components by the rotation R(θ). The arrows are never moved.
u·V = δ_ijuⁱV^j and ‖V‖ = √(V·V) agree in S and S′, because R leaves δ_ij unchanged.
u·V = ‖u‖ ‖V‖ cos φ, so u·V = 0 means the arrows are perpendicular in every frame.

Common mix-ups

"Turning the axes changes ‖V‖ or u·V." It changes the components only. Both panels agree at every θ.
"u·V = 0 means perpendicular in S only." Both frames report 0, so the arrows are perpendicular in both.
"Sliding the S′ origin changes V's components." It changes point coordinates. V is a difference of two points, and the shift cancels.

Things to try on your own

At stage 4, set θ to 90° and check the S′ rows against x′ = V_y, y′ = −V_x and the same rule for u.
Drag the tip of u to about (5, −1.25), a quarter longer. Predict the new u·V, then read both boxes.
At stage 3, set t = (2.0, −1.5) with the sliders and compare the tail and tip in S′ with their values at t = (0, 0).