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LIN-03

Dot product

2-D.

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Lineardot

Governing equation

mathbfucdotmathbfv=u1v1+u2v2\\mathbf{u}\\cdot\\mathbf{v}=u_1v_1+u_2v_2

u1∈[−10,10]u_1 \in \left[-10, 10\right]

where

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

The figure, part by part

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LIN-03 · gauge
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①

Governing law — red

mathbfucdotmathbfv=u1v1+u2v2\\mathbf{u}\\cdot\\mathbf{v}=u_1v_1+u_2v_2

u1∈[−10,10]u_1 \in \left[-10, 10\right]

u1=−10u_1 = -10u⋅v=−10u\cdot v = -10

②

Domain of the variable — blue

u_1_{\min} = -10,\quad u_1_{\max} = 10

u1∈[−10,10]u_1 \in \left[-10, 10\right]

u1=−10u_1 = -10u1=−10u_1 = -10

③

Response curve — green

mathbfucdotmathbfv=u1v1+u2v2\\mathbf{u}\\cdot\\mathbf{v}=u_1v_1+u_2v_2

u1∈[−10,10]u_1 \in \left[-10, 10\right]

u1=−10u_1 = -10u⋅v=−10u\cdot v = -10

④

Rate of change — amber

du⋅vdu1\frac{\mathrm{d}u\cdot v}{\mathrm{d}u_1}

u1∈[−10,10]u_1 \in \left[-10, 10\right]

u1=−9.874u_1 = -9.874u⋅v′=1u\cdot v^{\prime} = 1

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: u1u_1, u2u_2, v1v_1, v2v_2 →\rightarrow u⋅vu\cdot v. 2-D.

Purpose

Purpose: compute u⋅vu\cdot v from u1u_1, u2u_2, v1v_1, v2v_2 in Linear algebra via mathbfucdotmathbfv=u1v1+u2v2\\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 u1=3.000 —u_1 = 3.000\,\mathrm{—}, u2=4.000 —u_2 = 4.000\,\mathrm{—}, v1=1.000 —v_1 = 1.000\,\mathrm{—}, v2=0.000 —v_2 = 0.000\,\mathrm{—}, the governing relation mathbfucdotmathbfv=u1v1+u2v2\\mathbf{u}\\cdot\\mathbf{v}=u_1v_1+u_2v_2 yields u⋅v=3.000 —u\cdot v = 3.000\,\mathrm{—}. 2-D. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Outputs

  • dot u\cdot v3.000 —
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Narration of this film

2-D.

2-D.

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