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**Problem:** Given a vector \( \vec{v} \) and a real number \( \lambda \) (a scalar), prove that \( \|\lambda \cdot \vec{v}\| = |\lambda| \cdot \|\vec{v}\| \).

### Proof:

1. **Definition of the Norm of a Vector:**
   The norm (or magnitude) of a vector \( \vec{v} \) in a Euclidean space is defined as:
   \[
   \|\vec{v}\| = \sqrt{\vec{v} \cdot \vec{v}}
   \]
   where \( \vec{v} \cdot \vec{v} \) is the dot product of \( \vec{v} \) with itself.

2. **Scaling the Vector:**
   When the vector \( \vec{v} \) is scaled by a scalar \( \lambda \), the resulting vector is \( \lambda \cdot \vec{v} \).

3. **Norm of the Scaled Vector:**
   The norm of \( \lambda \cdot \vec{v} \) is given by:
   \[
   \|\lambda \cdot \vec{v}\| = \sqrt{(\lambda \cdot \vec{v}) \cdot (\lambda \cdot \vec{v})}
   \]

4. **Simplifying the Dot Product:**
   Using the properties of the dot product, specifically that \( (a \cdot \vec{u}) \cdot (b \cdot \vec{v}) = ab \cdot (\vec{u} \cdot \vec{v}) \) where \( a \) and \( b \) are scalars, we can simplify:
   \[
   (\lambda \cdot \vec{v}) \cdot (\lambda \cdot \vec{v}) = \lambda^2 \cdot (\vec{v} \cdot \vec{v})
   \]
   So the norm becomes:
   \[
   \|\lambda \cdot \vec{v}\| = \sqrt{\lambda^2 \cdot (\vec{v} \cdot \vec{v})}
   \]

5. **Further Simplification:**
   Since \( \lambda^2 \) is a scalar and can be factored out of the square root:
   \[
   \|\lambda \cdot \vec{v}\| = \sqrt{\lambda^2} \cdot \sqrt{\vec{v} \cdot \vec{v}}
   \]
   But \( \sqrt{\lambda^2} = |\lambda| \), so:
   \[
   \|\lambda \cdot \vec{v}\| = |\lambda| \cdot \|\vec{v}\|
   \]
   This completes the proof.

### Conclusion:
We have demonstrated that the norm of the scaled vector \( \lambda \cdot \vec{v} \) is equal to the absolute value of the scalar \( \lambda \) times the norm of the vector \( \vec{v} \). Hence, the statement \( \|\lambda \cdot \vec{v}\| = |\lambda| \cdot \|\vec{v}\| \) is true.

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Would you like further details or have any questions?

Here are some related questions you might find interesting:
1. How does the dot product relate to the angle between two vectors?
2. Can you demonstrate how the norm behaves under vector addition?
3. What properties of the dot product are used in this proof?
4. How does this proof change if \( \lambda \) is a complex number?
5. How does the triangle inequality relate to vector norms?

**Tip:** Always check the basic properties of operations like the dot product or vector scaling before starting proofs. These properties often provide the key steps in solving the problem.

Learn how to prove that the norm of a vector scaled by a scalar is equal to the absolute value of the scalar times the norm of the vector. This advanced high school-level problem explores vector norms, scalar multiplication, and dot products.

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