The Score Along the First Principal Direction

A robot arm carries three strain sensors $s_1, s_2, s_3$, each reading in centimetres. Over a long logging run the readings are mean-centred, and their covariance matrix (in $\text{cm}^2$) comes out as

$C = \begin{bmatrix} 15 & 0 & -6 \\ 0 & 9 & 6 \\ -6 & 6 & 12 \end{bmatrix}$

All three eigenvalues of $C$ are integers.

PCA takes the first principal direction $\mathbf{u}_1$ to be the unit eigenvector belonging to the largest eigenvalue of $C$. Such a unit eigenvector is only determined up to an overall sign, so this pipeline fixes the sign with the usual convention:

Sign convention. Orient $\mathbf{u}_1$ so that its first component is positive.

A fresh reading arrives, already mean-centred:

Its PC1 score is its coordinate along that first principal direction,

$z_1 = \mathbf{x} \cdot \mathbf{u}_1$

What is $z_1$?

Round your answer to two decimal places.