https://file.aiquickdraw.com/mathbot/file/17248551700055t028nnl.jpg

To find the length of vector \(\overrightarrow{BC}\), we'll use the coordinates of points \(B\) and \(C\) from the image. Let's start by identifying the coordinates:

- Point \(B\) has coordinates \((-5, -3)\).
- Point \(C\) has coordinates \((4, 2)\).

The formula for the length of a vector \(\overrightarrow{BC}\) is given by:

\[
|\overrightarrow{BC}| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

Where \((x_1, y_1)\) are the coordinates of point \(B\), and \((x_2, y_2)\) are the coordinates of point \(C\).

Substituting the values:

\[
|\overrightarrow{BC}| = \sqrt{(4 - (-5))^2 + (2 - (-3))^2}
\]
\[
|\overrightarrow{BC}| = \sqrt{(4 + 5)^2 + (2 + 3)^2}
\]
\[
|\overrightarrow{BC}| = \sqrt{9^2 + 5^2}
\]
\[
|\overrightarrow{BC}| = \sqrt{81 + 25}
\]
\[
|\overrightarrow{BC}| = \sqrt{106}
\]

Thus, the length of vector \(\overrightarrow{BC}\) is \(\sqrt{106}\), which is approximately 10.30 units.

Would you like further details on this calculation, or do you have any questions?

Here are 5 related questions:
1. What is the length of vector \(\overrightarrow{AB}\) in the same coordinate system?
2. How do you determine the direction of vector \(\overrightarrow{BC}\)?
3. Can you find the midpoint of segment \(BC\)?
4. How would the vector length change if \(B\) were moved to \((-5, 0)\)?
5. How can you use the vector components to find the angle between vectors \(\overrightarrow{AB}\) and \(\overrightarrow{BC}\)?

**Tip:** When finding the length of a vector, always double-check your coordinate points to ensure accuracy in your calculations.

Learn how to calculate the length of vector BC using the distance formula with coordinates (-5, -3) and (4, 2). This problem involves vector mathematics and coordinate geometry, suitable for high school students.

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