Question 5: Standard PCA has been performed on a centered dataset in …

Standard PCA has been performed on a centered dataset in $\mathbb{R}^3$. The first two principal components are given below:

$\mathbf{w}_1 = \frac{1}{\sqrt{3}} \cdot \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}, \quad \mathbf{w}_2 = \frac{1}{\sqrt{2}} \cdot \begin{bmatrix} 1 \\ -1 \\ 0 \end{bmatrix}$

Consider the data point in the dataset: $\begin{bmatrix} 2 & 1 & -1 \end{bmatrix}^T$. $(a, b)$ is the representation of this point in the coordinate system formed by the two principal components given above. The first and second coordinates correspond to PC-1 and PC-2 respectively.

Based on the above data, answer the given subquestions.

What is the value of a? Enter your answer correctly to three decimal places.