= Solution
For the sample, put $\bar x=n^{-1}\sum_ix_i$ and use the <sample covariance matrix>
$$
S=\frac1{n-1}\sum_{i=1}^{n}(x_i-\bar x)(x_i-\bar x)^T.
$$
Order its <eigenvalues> $\widehat\lambda_j$ decreasingly and choose unit orthogonal <eigenvectors> $\widehat\ell_j$. The <sample principal components> are the score columns
$$
\boxed{z_{ij}=\widehat\ell_j^T(x_i-\bar x).}
$$
Their <sample means> are zero and their sample <covariance> is diagonal, since $\widehat\ell_j^TS\widehat\ell_k=\widehat\lambda_j\mathbf1_{\{j=k\}}$. The $j$th <sample variance> is $\widehat\lambda_j$, and its <explained variance of a principal component> is $\widehat\lambda_j/\operatorname{tr}(S)$. Using divisor $n$ instead changes the numerical <variances> by a common factor but leaves the component directions unchanged.
Standardizing means replacing a variable with positive <sample variance> by $(x_{ij}-\bar x_j)/s_j$. This gives <principal component analysis on a correlation matrix>, using $C=D^{-1/2}SD^{-1/2}$ with $D=\operatorname{diag}(S)$. It removes arbitrary differences of measurement units and prevents a large-variance coordinate from dominating solely through its scale. It also changes the question being optimized: small-variance and possibly noisy coordinates receive equal marginal weight. <Principal component analysis> based on the <covariance matrix> can be more appropriate when the variables have common meaningful units and absolute variability matters. Standardization is therefore a substantive choice, not an automatic improvement. A zero-variance variable must first be removed. In the word-rating example the variables share the same numerical range, but their across-word dispersions can differ; standardization gives each attribute equal initial <variance>.
For the standardized eight-variable analysis, total <variance> is $\operatorname{tr}(C)=8$. The first three fractions of <explained variance of a principal component> are approximately \b[59.6%, 19.1% and 10.1%], respectively. The first two account for about \b[78.8%], and the first three for \b[88.9%]. The first direction is dominant but one direction alone loses substantial variation. Retaining two <principal components> is a plausible descriptive summary: their <eigenvalues> exceed $1$, whereas the third is below $1$. That cutoff is a heuristic, and the third <principal component> may still have interpretable content.
The reported coefficient rows are not unit-length vectors. The printed rounded rows have squared lengths $4.7514$, $1.5222$ and $0.8063$, consistent with their <eigenvalues> to the precision of the coefficient table. They are consistent with the scaled <principal component loading> convention $r_j=\sqrt{\lambda_j}\ell_j$. Such rows remain <eigenvectors>, but the score direction with unit length is $\ell_j=r_j/\sqrt{\lambda_j}$, or more accurately the row divided by its actual <Euclidean norm> when using rounded entries. Under this convention $r_{ij}$ is the <correlation> of standardized attribute $i$ with the variance-one score for component $j$:
$$
\operatorname{Corr}(Z_i,\ell_j^TZ/\sqrt{\lambda_j})=\sqrt{\lambda_j}\ell_{ij}.
$$
This explains both their sizes and why they must not be treated as the <unit vectors> from parts (i) and (ii).
The first <principal component> has positive coefficients on every attribute, strongest on friendliness, goodness, niceness, bravery and strength. With the favorable endpoints coded in the positive direction, it describes a broad favorable or impressive evaluation, with a potency contribution. The second contrasts movement and speed, with some size and strength, against the first three evaluative attributes; it is chiefly an activity dimension. The third is dominated by size, with smaller positive strength and negative movement/speed coefficients, distinguishing size or potency from activity. These are interpretations of directions of variation, not evidence of causal latent traits. Reversing a <principal component>'s overall sign changes neither its meaning as an axis nor its explained <variance>; endpoint coding fixes only how its positive side is described.
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