The answers for vectors $\mathbf{b_2}$ and $\mathbf{b_3}$ are:

$\mathbf{b_2} = \left[ \frac{149}{250}, -\frac{149}{200}, \frac{149}{500} \right]$

$\mathbf{b_3} = \left[ -\frac{1}{\sqrt{5}}, 0.2, \frac{2}{\sqrt{5}} \right]$

Let me know if you'd like further details or explanations!

Here are 5 related questions to deepen your understanding:

1. How does the Gram-Schmidt process ensure the orthogonality of the basis vectors?
2. What are the properties of an orthonormal basis in $\mathbb{R}^3$?
3. Why is it important to normalize the vectors during the Gram-Schmidt process?
4. How does the Gram-Schmidt process generalize to higher-dimensional spaces?
5. What is the significance of having an orthogonal matrix in linear algebra?

Tip: Always check if the computed basis vectors are orthogonal by calculating their dot product—it should be zero for orthogonal vectors.