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The image you uploaded presents a vector-related problem where you're asked to find a unit vector in the same direction as vector **\(\mathbf{v}\)**. The vector \(\mathbf{v}\) has the components given by the coordinates of two points, \(P(2, 3)\) and \(Q(-2, -6)\), as shown in the diagram.

### Steps to Solve:

1. **Find the Vector \(\mathbf{v}\):**
   The vector \(\mathbf{v}\) can be determined by subtracting the coordinates of point \(P\) from \(Q\):
   \[
   \mathbf{v} = Q - P = (-2 - 2, -6 - 3) = (-4, -9)
   \]

2. **Calculate the Magnitude of \(\mathbf{v}\):**
   The magnitude (or length) of vector \(\mathbf{v}\) is given by:
   \[
   |\mathbf{v}| = \sqrt{(-4)^2 + (-9)^2} = \sqrt{16 + 81} = \sqrt{97}
   \]

3. **Find the Unit Vector in the Direction of \(\mathbf{v}\):**
   A unit vector in the same direction as \(\mathbf{v}\) is obtained by dividing \(\mathbf{v}\) by its magnitude:
   \[
   \mathbf{u} = \frac{\mathbf{v}}{|\mathbf{v}|} = \left(\frac{-4}{\sqrt{97}}, \frac{-9}{\sqrt{97}}\right)
   \]

### Final Answer:
The unit vector in the direction of \(\mathbf{v}\) is:
\[
\mathbf{u} = \left(\frac{-4}{\sqrt{97}}, \frac{-9}{\sqrt{97}}\right)
\]

Would you like further details or have any questions?

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Here are five related questions to expand your understanding:

1. How would the unit vector change if the coordinates of points \(P\) and \(Q\) were different?
2. What is the significance of a unit vector in physics or engineering?
3. How would you find a unit vector if given a vector in three-dimensional space?
4. What role does the magnitude of a vector play in normalizing it to a unit vector?
5. Can a unit vector have negative components? Why or why not?

**Tip:** Always double-check the sign and magnitude calculations when dealing with vectors to avoid errors in the final unit vector.

The user is asked to find the unit vector in the direction of vector \(\mathbf{v}\) which is defined by the points P(2, 3) and Q(-2, -6).

Learn how to determine the unit vector in the direction of a vector defined by points P(2, 3) and Q(-2, -6). This problem includes key vector operations such as calculating the vector's magnitude and normalizing it to find the unit vector. Suitable for high school students in Grades 10-12.

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