A three-axis force sensor gives three output voltages

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Linear algebra and vector spaces · Linear algebra and vector spaces

Mathematical Methods for Physics, Ch Matrices, determinants and systems of linear equations

Problem

A three-axis force sensor gives three output voltages, but each channel picks up some of the force along its neighbouring axes. In calibration, a force of

1\,\mathrm{N}

along

x

alone gives readings

(2, 1, 0)\,\mathrm{mV}

on the three channels,

1\,\mathrm{N}

along

y

alone gives

(1, 2, 1)\,\mathrm{mV}

, and

1\,\mathrm{N}

along

z

alone gives

(0, 1, 2)\,\mathrm{mV}

. The sensor responds linearly, so the readings are

\mathbf{v} = \mathsf{C}\mathbf{F}

. ((a)) Write

\mathsf{C}

, with its unit, and say what its first column means. ((b)) Find

\mathsf{C}^{-1}

by elimination on

[\mathsf{C}\mid\mathsf{I}]

, and verify it. ((c)) Two loads give readings

(4, 8, 8)\,\mathrm{mV}

and

(2, 2, 2)\,\mathrm{mV}

. Find each force, and say why finding the inverse once was worth it here.

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