The triangle inequality for vectors states that for any two vectors $\vec{u}$ and $\vec{v}$, the following inequality holds:

$\| \vec{u} + \vec{v} \| \leq \| \vec{u} \| + \| \vec{v} \|$

where $\| \vec{u} \|$ denotes the magnitude (or length) of the vector $\vec{u}$.

To verify this inequality, we will use the properties of vector magnitudes and the Cauchy-Schwarz inequality. Here's a step-by-step approach:

Step 1: Expand $\| \vec{u} + \vec{v} \|^2$

We start by squaring both sides of the inequality to make it easier to manipulate. The left side is the square of the magnitude of $\vec{u} + \vec{v}$:

$\| \vec{u} + \vec{v} \|^2 = (\vec{u} + \vec{v}) \cdot (\vec{u} + \vec{v})$

Using the distributive property of the dot product, we get:

$\| \vec{u} + \vec{v} \|^2 = \vec{u} \cdot \vec{u} + 2 \vec{u} \cdot \vec{v} + \vec{v} \cdot \vec{v}$

Since $\vec{u} \cdot \vec{u} = \| \vec{u} \|^2$ and $\vec{v} \cdot \vec{v} = \| \vec{v} \|^2$, we have:

$\| \vec{u} + \vec{v} \|^2 = \| \vec{u} \|^2 + 2 \vec{u} \cdot \vec{v} + \| \vec{v} \|^2$

Step 2: Apply the Cauchy-Schwarz inequality

The Cauchy-Schwarz inequality states that for any two vectors $\vec{u}$ and $\vec{v}$,

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

Thus, we can bound the middle term in our expanded expression:

$2 \vec{u} \cdot \vec{v} \leq 2 \| \vec{u} \| \| \vec{v} \|$

Therefore,

$\| \vec{u} + \vec{v} \|^2 \leq \| \vec{u} \|^2 + 2 \| \vec{u} \| \| \vec{v} \| + \| \vec{v} \|^2$

Step 3: Recognize the right-hand side as a perfect square

The right-hand side is a perfect square:

$\| \vec{u} \|^2 + 2 \| \vec{u} \| \| \vec{v} \| + \| \vec{v} \|^2 = (\| \vec{u} \| + \| \vec{v} \|)^2$

Thus, we have:

$\| \vec{u} + \vec{v} \|^2 \leq (\| \vec{u} \| + \| \vec{v} \|)^2$

Step 4: Take square roots

Taking the square root of both sides gives the desired result:

$\| \vec{u} + \vec{v} \| \leq \| \vec{u} \| + \| \vec{v} \|$

This confirms that the triangle inequality holds for vectors.

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Would you like a more detailed explanation or examples of how to apply this? Here are some related questions for further study:

1. How does the triangle inequality relate to geometric interpretation in Euclidean space?
2. Can you prove the reverse triangle inequality $\| \vec{u} - \vec{v} \| \geq | \| \vec{u} \| - \| \vec{v} \| |$?
3. How do you use the triangle inequality in vector space norms?
4. What is the role of the Cauchy-Schwarz inequality in vector spaces?
5. How does the triangle inequality extend to other types of norms (e.g., Manhattan or Minkowski)?

Tip: The triangle inequality also holds for any norm, not just the Euclidean norm. For example, in the space of continuous functions, it holds for the supremum norm!