To find the length of a vector, we use the formula for the Euclidean norm (or magnitude) of a vector. For a vector $\mathbf{v} = \begin{pmatrix} a \\ b \\ c \end{pmatrix}$, the length is given by:

$\text{Length of } \mathbf{v} = \sqrt{a^2 + b^2 + c^2}$

In this case, the vector is:

$\mathbf{v} = \begin{pmatrix} 2 \\ -5 \\ 7 \end{pmatrix}$

Let's calculate its length.

$\text{Length} = \sqrt{2^2 + (-5)^2 + 7^2} = \sqrt{4 + 25 + 49} = \sqrt{78}$

Thus, the length of the vector is:

$\sqrt{78} \approx 8.83$

Would you like any further explanation on this, or should I clarify any step?

Here are 5 related questions to expand your understanding:

1. How does the Euclidean norm differ from other types of vector norms?
2. What is the significance of a vector's length in physics or engineering applications?
3. Can you calculate the length of a 2D vector using a similar formula?
4. How would the length change if the vector was scaled by a factor of 2?
5. How do you interpret the direction of a vector when you have its components?

Tip: Normalizing a vector involves dividing each of its components by its length, which results in a unit vector pointing in the same direction.