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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.

What is the length of the vector (2, -5, 7)?

Learn how to calculate the length of the 3D vector (2, -5, 7) using the Euclidean norm formula. This problem involves finding the magnitude of a vector in three dimensions, suitable for high school students in Grades 10-12.

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