Scalars vs. Vectors Under Rotation

Source

Lecture: 4 — Scalar invariants
Simulation: Simulation 1 — lecture visual aid (lecture-demo mode)
Concept: Vector components change under a rotation of Cartesian axes, but scalar quantities — each vector's norm and the norm of their difference — remain invariant.
Serves: Identify rotations as coordinate transformations between Cartesian systems; use the Euclidean metric to define invariants; distinguish scalars from vectors under coordinate transformations; derive the spatial distance (interval) as a scalar invariant.
Inquiry: What changes when the axes rotate, and what stays the same? Is the distance between two vector tips a scalar?

Pedagogy principles applied

ID	Principle	How it shows up here
A1	One concept	Only scalar invariance of norms/distance under rotation — no second concept.
A2	No decoration	Drop-lines, dashed A−B vector and readout rows all serve the change-vs-invariant comparison.
A3/A4	Formal off by default	Rotation equations & Euclidean metric live in The Physics revision sheet, which is closed on load.
B1	Standard terms	rotation, components, norm, scalar, coordinate axes — no invented jargon.
B3	Define symbols	Aₓ, ‖A‖, θ defined on first appearance; S vs S′ kept visually distinct.
B4	Descriptive headers	"Original axes (S)" / "Rotated axes (S′)".
C1	No orphan numbers	Every readout is referenced by the staged narrative and the prediction.
C2	Identity + frame	Each readout carries its symbol and frame label (S / S′).
D1	Panels dominate	Two coordinate panels fill the stage; readouts sit compactly beneath.
D2	Readable text	Axis/vector labels ≥12px; readout numbers 14px.
D3	Arrows not dots	A, B and A−B are coloured, labelled arrows.
D4	No sub-scroll	Both panels and readouts fit without scrolling.
D5	No overlap	Labels are bounds-checked at all θ and tip positions.
D8	Labels only on canvas	Explanations live in side panels, not painted on the canvas.
J1	Consistent notation	bold A, subscripts, prime for S′, θ for the angle throughout.
J3	Correct rotation	Passive rotation matrix applied correctly (R^T(θ)); norms computed independently in each frame to demonstrate, not enforce, equality.

Teach me

Speed

SOriginal axes (S)axes x, y

drag the tip of A or B

Readouts · S frame

invariant

interval

S′Rotated axes (S′)θ = 0°

Readouts · S′ frame

invariant

interval

Guided Inquiry

1 · Two arrows, two sets of axes

Let’s meet our two arrows, A and B, nailed firmly to the plane. The left panel measures them against the axes x and y; the right panel measures the same two arrows against axes x′ and y′, turned through an angle θ. At θ = 0° the two descriptions coincide.

2 · Speed and velocity

First, a warm-up from everyday life. One car drives north at 60 km/h, and a second car drives east at 60 km/h. What do you think the two cars have in common?

3 · Turning the axes

Now back to our arrows. They stay nailed where they are while the primed axes are turned through 90°. Compared with the unprimed frame, what do you think happens to the two components of A, and to the length of A?

4 · Why the length survives

Time to dig into why. At θ = 90° the primed components of A read 4.00 and −3.00 against 3.00 and 4.00 unprimed, yet both frames report ‖A‖ = 5.00. What do you think holds that agreement together?

5 · Which lengths are scalars?

Now stretch the idea a little further. ‖A‖ and ‖B‖ are both measured out from the origin, which the two frames share. But the arrow running from B’s tip to A’s tip starts nowhere near that origin. Which lengths do you think come out the same in both frames?

6 · Scalars, and what comes next

Let’s put a name to it. A number every frame reports identically is a scalar: ‖A‖ = 5.00, ‖B‖ = 3.61 and ‖A−B‖ = 5.10. A_x and A_y aren’t — they’re just one frame’s description of an arrow that never moved.

One subtlety: invariant isn’t the same as constant. Drag a tip and every number moves, but the frames move together. Relativity keeps this very pattern with a new metric — t and x are frame-dependent, but s² = −(ct)² + x² is not.

Listen

Controls

▾

Rotation angle θ θ = 0°

0°360°

Layers

Rotation of axes (S → S′, axes rotated CCW by θ)

Matrix form (passive rotation)

Euclidean norm — a scalar

Distance (spatial interval) — invariant

The Physics

Scalars vs. vectors under rotation · what this sim shows, the physics behind it, and every control

At a glance

The sim puts two arrows, A and B, in a plane and reads them off two pairs of axes: S, and S′, turned by an angle θ about the same origin. Physical laws cannot depend on the choice of axes, so it matters which numbers change when the axes turn.

What you'll learn

How the numbers that describe an arrow depend on the axes, and the rule that links two sets of axes.
How the length of an arrow, and the distance between two tips, depend on θ.
What "scalar" means, and where the idea returns in spacetime.

The sim's two panels: the same arrows, over S axes (left) and turned S′ axes (right).

What's on screen

Arrow A (solid blue): drawn from the origin, same in both panels.
Arrow B (solid amber): same in both panels.
Dashed drop-lines (layer ①): from each tip, perpendicular to the panel's own axes. The feet are the components.
Grey axes: x, y in the left panel (S); x′, y′ in the right panel (S′), turned counterclockwise by θ.
Green dashed arrow (layer ③): A − B, drawn from the tip of B to the tip of A.
Readouts under each panel: the components in that panel's axes and, with ② and ③, the lengths.

See itStart the sweep, with both arrows and their drop-lines on. Watch for: the axes in the right panel turning while A and B stay where they are.

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 counterclockwise about the origin. The arrows do not move. Dragging it pauses the sweep.	0°–360°, step 1° opens at 0°
① Components	Shows arrows A and B, their dashed drop-lines, and the component rows under each panel.	on / off opens on
② Norms	Adds one length row per arrow, ‖A‖ and ‖B‖, under each panel.	on / off opens off
③ ‖A−B‖	Adds the green dashed arrow from B to A, and a row with its length.	on / off opens off
Drag the tips	Moves A or B. Drag in the left panel (S); both panels follow.	each coordinate within ±5.8 opens at A = (3, 4), B = (−2, 3)
Readouts	The numbers behind each panel, to two decimals: S on the left, S′ on the right.	two panels
⏸ Pause / ▶ Play	Runs or freezes the sweep. While running, θ grows by 18° per second at 1× (a full turn in 20 s) and wraps to 0°.	top bar
Speed	Scales the sweep rate.	0.25×–4× opens at 1×
↻ Reset	Puts A, B and θ back to their opening values and restarts the sweep.	top bar

Fixed: both frames share one origin and one length unit, so only rotation is involved. The plane is two-dimensional.

Components belong to the axes

An arrow's components are where perpendicular drop-lines from its tip meet a chosen pair of axes. In S′ the axes are the S axes turned counterclockwise by θ, and the arrows stay put (a passive rotation). The new components follow from the old:

Components in S′ from components in S: the rule the sim uses.

For A = (3, 4) and θ = 30°, S′ reads A = (4.60, 1.96):

One arrow, two pairs of axes, turned by 30°.

Key idea. The same arrow has different components relative to different axes. Components belong to the arrow together with the axes, not to the arrow alone.

See itSweep the axes through a full turn. Watch for: the S′ numbers changing continuously while the S numbers and both arrows stay fixed.

The length is shared out between the components

Square both S′ components of A and add them. The cross terms, ±2A_xA_y sin θ cos θ, cancel and cos²θ + sin²θ = 1, leaving A_x′² + A_y′² = A_x² + A_y² for every θ.

The Euclidean norm, from S components and from S′ components.

This is the sim's own θ slider. Numbers are rounded to two decimals.

Turning the axes moves length from one component to the other, and the sum of squares stays the same. The sim computes the S′ length from the S′ components, separately from the S length.

See itTurn the axes until x′ points along A, with the norm rows on. Watch for: nearly all of A in A_x′ and almost none in A_y′, yet both ‖A‖ rows agree.

The distance between two points is a scalar

Components subtract, so A − B is an arrow too, and it turns with A and B. Its length is the distance d between the two tips, and it needs no reference to the origin.

The ③ layer: the separation of the tips, measured in each frame separately.

Scalar (invariant). A quantity with the same value in S and S′, whatever θ is. ‖A‖, ‖B‖ and ‖A−B‖ are all scalars. The components are not.

Invariant does not mean constant. Drag a tip and ‖A−B‖ changes, in both panels together. At the opening positions A − B = (5, 1) and d = √26 ≈ 5.10.

See itTurn on all three layers and sweep. Drag a tip to move the arrows. Watch for: the two ‖A−B‖ rows matching while θ keeps turning.

Where it matters

Speed and velocity. A particle's speed is the length of its velocity vector, v² = V_x² + V_y². Two observers with rotated axes disagree on the components of V but agree on v.
Laws are built from scalars. Different observers use different axes, so a law is written with quantities they can all agree on, such as lengths and inner products.
Spacetime. Special relativity keeps this construction and changes the rule for length. Its interval is Δs² = −c²Δt² + Δx² + Δy² + Δz². The sim shows only rotations of space, not this interval.

Go deeper: the metric, and the same idea with a boost

The inner product uses the table δ_ij. Cartesian to Cartesian leaves its entries unchanged.

The table δ_ij is not a scalar: it keeps its entries only when one Cartesian system is exchanged for another. The scalar is the number U·V it produces.

The spacetime interval: a time term plus the Euclidean part.

A rotation keeps the components on a circle; a boost keeps an event's (x, ct) on a hyperbola. In 1+1 dimensions the interval is x² − (ct)².

Summary

In one line. Turning the axes changes an arrow's components but not its length, nor the distance between two tips: those are scalars.

Key points

Components belong to an arrow and a pair of axes: A_x′ = A_x cos θ + A_y sin θ, A_y′ = −A_x sin θ + A_y cos θ.
A_x² + A_y² is the same in S and S′, so ‖A‖ is a scalar.
A − B turns like any arrow, so the distance ‖A−B‖ is a scalar too.

Common mix-ups

"Invariant means constant." Drag a tip and ‖A‖ changes. Invariant means S and S′ agree at every θ.
"Turning the axes just swaps the components." Only at 90° (and then with a sign: (3, 4) becomes (4, −3)). At 53° the same A reads (5.00, 0.01).
"Only lengths from the origin are scalars." ‖A−B‖ does not touch the origin, and its two rows still match.

Things to try on your own

Set θ to 90°, compare A and B in S′ with A and B in S, and write down the rule.
Turn on ②, drag a tip around the left panel, and compare the two ‖A‖ rows at several positions.
Turn on ③ and sweep a full turn. List which rows change and which stay fixed.