LIN-05

Projection length

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Linearproj

Governing equation

|\\mathrm{proj}_v u|=|u\\cdot\\hat v|

$\theta \in \left[0, 180\right]$

where

|u|: |u| (—)
\theta: θ (°)
|proj|: proj (—)

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

|\\mathrm{proj}_v u|=|u\\cdot\\hat v|

$\theta \in \left[0, 180\right]$

$\theta = 0$ $|proj| = 5$

Along the curve

②

Domain of the variable — blue

\theta_{\min} = 0,\quad \theta_{\max} = 180

$\theta \in \left[0, 180\right]$

$\theta = 0$ $\theta = 0$

Along the curve

③

Response curve — green

|\\mathrm{proj}_v u|=|u\\cdot\\hat v|

$\theta \in \left[0, 180\right]$

$\theta = 0$ $|proj| = 5$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}|proj|}{\mathrm{d}\theta}

$\theta \in \left[0, 180\right]$

$\theta = 1.132$ $|proj|^{\prime} = -0.002$

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-05 — Projection length) is the form associated with proj. Working symbols: $|u|$ , $\theta$ $\rightarrow$ $|proj|$ . Onto a line.

Purpose

Purpose: compute

|proj|

from

|u|

\theta

in Linear algebra via

|\\mathrm{proj}_v u|=|u\\cdot\\hat v|

Onto a line. 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

|u| = 5.000\,\mathrm{—}

\theta = 40.000\,\mathrm{^{\circ}}

, the governing relation

|\\mathrm{proj}_v u|=|u\\cdot\\hat v|

yields

|proj| = 3.830\,\mathrm{—}

. Onto a line. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

|u| |u|5 —θ \theta40 °

Outputs

proj |proj|3.830 —

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Narration of this film

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