https://r2.erweima.ai/mathbot/image/1726287756252rfsicjzg.jpg

The image presents a vector problem involving two points \( A = (5, -7) \) and \( B = (2, -9) \) in \( \mathbb{R}^2 \). The task is to find the position vector of \( \overrightarrow{AB} \) in standard position, which corresponds to a point \( P \).

### Solution:

The position vector \( \overrightarrow{AB} \) is calculated by subtracting the coordinates of point \( A \) from the coordinates of point \( B \):

\[
\overrightarrow{AB} = B - A = (x_B - x_A, y_B - y_A)
\]

Where \( A = (5, -7) \) and \( B = (2, -9) \):

\[
\overrightarrow{AB} = (2 - 5, -9 - (-7)) = (-3, -2)
\]

Thus, the point \( P \), which represents the position vector \( \overrightarrow{AB} \) in standard position, is:

\[
P = (-3, -2)
\]

### Final Answer:
\[
P = (-3, -2)
\]

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Would you like further details or explanations?

Here are some related questions for practice:
1. How do you find the magnitude of vector \( \overrightarrow{AB} \)?
2. What would the unit vector of \( \overrightarrow{AB} \) be?
3. If \( C = (4, 1) \), what is the vector \( \overrightarrow{AC} \)?
4. How can you check if two vectors are parallel?
5. What is the dot product of vectors \( \overrightarrow{AB} \) and \( \overrightarrow{AC} \)?

**Tip:** When subtracting coordinates to find a vector, remember to subtract the corresponding components, not just the magnitudes.

Let A = (5, -7) and B = (2, -9) be points in R^2. Then in standard position the vector AB is the position vector of which point P?

Solve the vector problem by finding the position vector of AB, where A = (5, -7) and B = (2, -9). Learn vector subtraction and its application in coordinate geometry. Suitable for high school students in Grades 9-12.

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