The Inner Product in a Rotated Frame

Source

Collection: Wiki SR simulations (v1) — SuperSTEM Special-Relativity engine sim.
Concept: The Euclidean inner product A·B = A_xB_x + A_yB_y is invariant under a frame rotation — the Euclidean warm-up for the invariance of the Minkowski inner product under a Lorentz boost.
Serves: Geometric intuition that a frame change only relabels components: rotating the S′ grid changes each component, yet A·B and the magnitudes |A|, |B| are untouched for every angle θ.
Inquiry: What happens to A_x′B_x′ + A_y′B_y′ as θ sweeps? Why does the combination stay fixed while its parts change? How does this map onto a Lorentz boost?
Misconception: That the inner product oscillates, shrinks, or flips sign as the axes turn — i.e. reading a frame relabelling as a change to the vectors themselves.

Pedagogy principles applied

ID	Principle	How it shows up here
H1	Misconception is structural, not a text card	The "oscillates", "shrinks" and "flips sign" errors are wrong prediction choices; each rebuttal appears only if the student picks it.
D6/D7	Guided inquiry reveals in stages	Four sidebar step-cards run fundamental → full: predict → rotate and observe → verify → the Minkowski payoff.
J	Invariant panels are scoped to invariance sims	This sim's lesson IS invariance, so the A·B / A′·B′ invariant readout is shown — the two frames are compared side by side.
A2	Fewest objects	Opens with two arrows and two frames; no vestigial metric cards or counts that no step reads.
Physics honesty	No false animation	θ is a static slider the student sets — the frame rotation is a re-labelling of axes, not invented motion over time; the auto-sweep was dropped.

Speed

S — lab frame

vector A vector B

S′ — rotated frame (locked)

S′ axes / grid A components

Guided Inquiry

1 · Predict the inner product

Rotate the frame S′ by an angle θ. The measured components (A_x′, A_y′) and (B_x′, B_y′) all change with θ. What does the inner product built from them, A_x′B_x′ + A_y′B_y′, do?

2 · Rotate the frame

Drag the θ slider in the sidebar. Watch the right panel: the grid and axes of S′ turn, the arrows A and B stay pinned. The dashed lines show A's components on the current axes — they clearly change with θ.

3 · What did A·B do?

As you swept θ through the whole family of rotations, the two gold rows — A·B in S and A′·B′ in S′ — …

4 · The Minkowski payoff

This is the Euclidean warm-up for spacetime. A rotation preserves x² + y² and the dot product A·B. A Lorentz boost is the same idea with one sign flipped: it preserves (ct)² − x² and the Minkowski inner product A_tB_t − A_xB_x. A frame change reshuffles components but preserves the metric combination. Turn on ∑ Formal for the algebra, or drag the arrow tips to test any A and B. Hit Finish to explore freely.

Controls

▾

frame angle θ = 0°

0°360°

Invariants

▾

A·B (frame S)4.000

A′·B′ (frame S′)4.000

|A| = |A′|2.236

|B| = |B′|2.236

Components in the rotated frame

Length is invariant

Inner product is invariant

Minkowski analogue (a boost)

A rotation preserves x² + y² and the dot product; a Lorentz boost preserves (ct)² − x² and the Minkowski inner product. Same logic — a frame change reshuffles components but preserves the metric combination — with a minus sign in the metric.