LIN-24

Vector cosine

Planar direction cosine.

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GoverningVector cosine

Governing equation

\\cos\\theta=(u\\cdot v)/(|u||v|)

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

where

ux: ux (—)
uy: uy (—)
vx: vx (—)
vy: vy (—)
c: Vector cosine (—)

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

\\cos\\theta=(u\\cdot v)/(|u||v|)

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

$ux = -20$ $c = -0.981$

Along the curve

②

Domain of the variable — blue

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

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

$ux = -20$ $ux = -20$

Along the curve

③

Response curve — green

\\cos\\theta=(u\\cdot v)/(|u||v|)

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

$ux = -20$ $c = -0.981$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}c}{\mathrm{d}ux}

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

$ux = -19.75$ $c^{\prime} = 0.002$

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-24 — Vector cosine) is the form associated with Vector cosine. Working symbols: $ux$ , $uy$ , $vx$ , $vy$ $\rightarrow$ $c$ . Planar direction cosine. Pedagogical SI sheet with a live model and a swept parameter.

Purpose

Purpose: compute

c

from

ux

uy

vx

vy

in Linear algebra via

\\cos\\theta=(u\\cdot v)/(|u||v|)

Planar direction cosine. 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 = 4.000\,\mathrm{—}

vx = 1.000\,\mathrm{—}

vy = 0.000\,\mathrm{—}

, the governing relation

\\cos\\theta=(u\\cdot v)/(|u||v|)

yields

c = 0.600\,\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 uy4 —vx vx1 —vy vy0 —

Outputs

Vector cosine c0.600 —

Reading speed

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Narration of this film

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

Planar direction cosine. Pedagogical SI sheet with a live model and a swept parameter.

Reading speed

Watch on YouTube