What Two Patterns Cannot Explain

A four-channel sensor produces one reading per channel, stacked into a vector

$\mathbf{v} = \begin{bmatrix} 5 \\ 1 \\ 3 \\ 4 \end{bmatrix}$

An engineer models the signal as a combination of two known patterns, $\mathbf{w}_1$ and $\mathbf{w}_2$ , which span a plane $W$ inside $\mathbb{R}^4$ . One entry of the second pattern was lost, and is written as $t$ :

Channel	$1$	$2$	$3$	$4$
$\mathbf{w}_1$	$2$	$1$	$0$	$2$
$\mathbf{w}_2$	$1$	$t$	$2$	$0$
Reading $\mathbf{v}$	$5$	$1$	$3$	$4$

The design guarantee is that the two patterns are orthogonal to each other, and that fact alone pins down $t$ .

Recover $t$ , then project $\mathbf{v}$ onto the plane $W$ . The part of the reading that no combination of the two patterns can reproduce is $\mathbf{v} - \text{proj}_W \mathbf{v}$ .

What is the length of that unexplained part? Round your answer to 2 decimal places.

What Two Patterns Cannot Explain

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