The Calibration Value That Collapses the Array

A three-sensor array is tested with one standard pulse. Each sensor answers with a three-channel reading, so each sensor's response is a vector in $\mathbb{R}^3$ :

	Channel 1	Channel 2	Channel 3
Sensor 1 ( $v_1$ )	$2$	$-1$	$3$
Sensor 2 ( $v_2$ )	$1$	$4$	$-2$
Sensor 3 ( $v_3$ )	$t$	$5$	$4$

Only sensor 3's channel-1 gain is adjustable; it is set by a calibration constant $t$ , which may be any real number.

$v_1 = \begin{pmatrix} 2 \\ -1 \\ 3 \end{pmatrix}, \qquad v_2 = \begin{pmatrix} 1 \\ 4 \\ -2 \end{pmatrix}, \qquad v_3 = \begin{pmatrix} t \\ 5 \\ 4 \end{pmatrix}$

The engineers call the array degenerate when the three response vectors $\{v_1, v_2, v_3\}$ are linearly dependent — at that setting one sensor is reporting nothing the other two do not already determine, and the array has quietly lost a dimension of coverage.

Exactly one value of $t$ makes the array degenerate.

Find that value of $t$ . Round your answer to 2 decimal places.

The Calibration Value That Collapses the Array

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