The question asks when equality holds in the Cauchy-Schwarz inequality, which states:

$|\mathbf{u} \cdot \mathbf{v}| \leq \|\mathbf{u}\| \|\mathbf{v}\|$

Equality holds in the Cauchy-Schwarz inequality if and only if one vector is a scalar multiple of the other. In other words, if $\mathbf{u}$ and $\mathbf{v}$ are linearly dependent, then:

$|\mathbf{u} \cdot \mathbf{v}| = \|\mathbf{u}\| \|\mathbf{v}\|$

Answer Choices Analysis:

1. When one vector is a scalar multiple of the other.
   This is correct. Equality in the Cauchy-Schwarz inequality occurs when $\mathbf{u} = \lambda \mathbf{v}$ for some scalar $\lambda$.

2. When $\mathbf{u}$ and $\mathbf{v}$ are orthogonal.
   This is incorrect. When vectors are orthogonal, their dot product is zero, so there is no equality condition in the form given by the Cauchy-Schwarz inequality.

3. When $\mathbf{u}, \mathbf{v}$ are vectors in $\mathbb{R}^n$ for any $n$.
   This is a general statement about the applicability of the inequality and does not specifically describe when equality holds.

4. When $\mathbf{u} = 0$ or $\mathbf{v} = 0$.
   This is correct. If either vector is zero, then both sides of the inequality are zero, so equality holds trivially.

5. Equality always holds.
   This is incorrect, as equality does not always hold in the Cauchy-Schwarz inequality. It only holds under specific conditions, as mentioned.

Correct Answers:

• When one vector is a scalar multiple of the other.
• When $\mathbf{u} = 0$ or $\mathbf{v} = 0$.

Would you like a further explanation on the Cauchy-Schwarz inequality or any other details?

Related Questions

1. What is the general form of the Cauchy-Schwarz inequality in different vector spaces?
2. How can we use the Cauchy-Schwarz inequality to prove other inequalities?
3. Why does equality only hold when one vector is a scalar multiple of the other?
4. How is the Cauchy-Schwarz inequality applied in complex vector spaces?
5. What are some real-world applications of the Cauchy-Schwarz inequality?

Tip: Remember that inequalities often describe bounds, so equality conditions are usually special cases that describe when the bound is tight.