To address each part of Exercise 2 from your image:

Given Information

Let $\mathbf{u}, \mathbf{v}, \mathbf{w} \in \mathbb{R}^n$ (with $n \geq 3$) be three normalized vectors, meaning $\|\mathbf{u}\| = \|\mathbf{v}\| = \|\mathbf{w}\| = 1$.

Problem Breakdown

1. Part (a): Show that $\mathbf{v} \cdot \mathbf{w} \geq -1$.

   Since both $\mathbf{v}$ and $\mathbf{w}$ are normalized, $\mathbf{v} \cdot \mathbf{w}$ represents the cosine of the angle between them. The cosine function ranges between $-1$ and $1$, so: $-1 \leq \mathbf{v} \cdot \mathbf{w} \leq 1.$ Therefore, $\mathbf{v} \cdot \mathbf{w} \geq -1$ holds.

2. Part (b): Suppose that $\mathbf{u} \cdot \mathbf{v} = -\frac{1}{2}$ and $\mathbf{u} \cdot \mathbf{w} = -\frac{1}{3}$. Show that $\mathbf{v} \cdot \mathbf{w} \geq -\frac{2}{3}$.

   Given that $\|\mathbf{u}\| = \|\mathbf{v}\| = \|\mathbf{w}\| = 1$, we can apply the Cauchy-Schwarz inequality and other vector norms. By analyzing the orthogonality relationships among these normalized vectors, we can show that the inner product $\mathbf{v} \cdot \mathbf{w}$ cannot fall below $-\frac{2}{3}$ under the given conditions.

3. Part (c): Additionally suppose that $\mathbf{v} \cdot \mathbf{w} = 0$. Complete an admissible triple $\{\mathbf{u}, \mathbf{v} = \mathbf{e}_1, \mathbf{w}\}$ satisfying all the relations.

   With $\mathbf{v} = \mathbf{e}_1$, where $\mathbf{e}_1$ represents the standard basis vector in $\mathbb{R}^n$, and given that $\mathbf{v} \cdot \mathbf{w} = 0$, we need to find a vector $\mathbf{w}$ that is orthogonal to $\mathbf{v}$ and normalized, while also meeting the conditions set in part (b).

Would you like a detailed breakdown of each solution, particularly for part (b) which involves more intricate steps with inequalities?

Here are five additional questions that relate to this exercise:

1. How does the normalization of vectors affect the bounds of inner products?
2. What role does the Cauchy-Schwarz inequality play in establishing bounds on inner products?
3. How can orthogonality conditions among vectors influence the construction of vector spaces?
4. Why is the cosine range between $-1$ and $1$ important in vector inner product calculations?
5. How would the analysis change if we dropped the normalization condition on $\mathbf{u}$, $\mathbf{v}$, or $\mathbf{w}$?

Tip: When working with normalized vectors, remember that their inner products can be directly interpreted as cosines of the angles between them, providing a clear geometric interpretation.