LIN-25

Parallelogram area

2-D cross magnitude.

Reading speed

GoverningParallelogram area

Governing equation

A=|u_x v_y-u_y v_x|

$vy \in \left[-20, 20\right]$

where

ux: ux (—)
uy: uy (—)
vx: vx (—)
vy: vy (—)
A: Parallelogram area (—)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

A=|u_x v_y-u_y v_x|

$vy \in \left[-20, 20\right]$

$vy = -20$ $A = 60$

Along the curve

②

Domain of the variable — blue

vy_{\min} = -20,\quad vy_{\max} = 20

$vy \in \left[-20, 20\right]$

$vy = -20$ $vy = -20$

Along the curve

③

Response curve — green

A=|u_x v_y-u_y v_x|

$vy \in \left[-20, 20\right]$

$vy = -20$ $A = 60$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}A}{\mathrm{d}vy}

$vy \in \left[-20, 20\right]$

$vy = -19.75$ $A^{\prime} = -3$

Along the curve

Lecture brief

Historical brief

Determinants, eigenvalues, least squares and Cayley–Hamilton are the nineteenth-century matrix craft behind every coupled system. The sheets invert, project and diagonalise small n. This sheet (LIN-25 — Parallelogram area) is the form associated with Parallelogram area. Working symbols: $ux$ , $uy$ , $vx$ , $vy$ $\rightarrow$ $A$ . 2-D cross magnitude. Pedagogical SI sheet with a live model and a swept parameter.

Purpose

Purpose: compute

A

from

ux

uy

vx

vy

in Linear algebra via

A=|u_x v_y-u_y v_x|

2-D cross magnitude. Use it when a real linear algebra question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

ux = 3.000\,\mathrm{—}

uy = 0.000\,\mathrm{—}

vx = 1.000\,\mathrm{—}

vy = 2.000\,\mathrm{—}

, the governing relation

A=|u_x v_y-u_y v_x|

yields

A = 6.000\,\mathrm{—}

. One governing identity, SI units, a single sweep on the sheet. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

ux ux3 —uy uy0 —vx vx1 —vy vy2 —

Outputs

Parallelogram area A6.000 —

Reading speed

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Narration of this film

One governing identity, SI units, a single sweep on the sheet.

2-D cross magnitude. Pedagogical SI sheet with a live model and a swept parameter.

Reading speed

Watch on YouTube