LIN-03

Dot product

2-D.

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Lineardot

Governing equation

\\mathbf{u}\\cdot\\mathbf{v}=u_1v_1+u_2v_2

where

u_1: u1 (—)
u_2: u2 (—)
v_1: v1 (—)
v_2: v2 (—)
u\cdot v: dot (—)

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-03 — Dot product) is the form associated with dot. Working symbols: $u_1$ , $u_2$ , $v_1$ , $v_2$ $\rightarrow$ $u\cdot v$ . 2-D.

Purpose

Purpose: compute

u\cdot v

from

u_1

u_2

v_1

v_2

in Linear algebra via

\\mathbf{u}\\cdot\\mathbf{v}=u_1v_1+u_2v_2

2-D. 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_1 = 3.000\,\mathrm{—}

u_2 = 4.000\,\mathrm{—}

v_1 = 1.000\,\mathrm{—}

v_2 = 0.000\,\mathrm{—}

, the governing relation

\\mathbf{u}\\cdot\\mathbf{v}=u_1v_1+u_2v_2

yields

u\cdot v = 3.000\,\mathrm{—}

. 2-D. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

u1 u_13 —u2 u_24 —v1 v_11 —v2 v_20 —

Outputs

dot u\cdot v3.000 —

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Narration of this film

2-D.

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