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To find the components of the vector \( \overrightarrow{P_1P_2} \) where \( P_1(4, 1, -3) \) and \( P_2(4, 5, -3) \), we need to subtract the coordinates of \( P_1 \) from \( P_2 \).

The general formula for the vector between two points \( P_1(x_1, y_1, z_1) \) and \( P_2(x_2, y_2, z_2) \) is:

\[
\overrightarrow{P_1P_2} = (x_2 - x_1, y_2 - y_1, z_2 - z_1)
\]

Substituting the coordinates of \( P_1(4, 1, -3) \) and \( P_2(4, 5, -3) \):

\[
\overrightarrow{P_1P_2} = (4 - 4, 5 - 1, -3 - (-3))
\]

\[
\overrightarrow{P_1P_2} = (0, 4, 0)
\]

So, the components of the vector \( \overrightarrow{P_1P_2} \) are \( (0, 4, 0) \).

Would you like more details or have any questions?

Here are 5 related questions for further exploration:
1. What is the magnitude of the vector \( \overrightarrow{P_1P_2} \)?
2. How do we find the direction of the vector \( \overrightarrow{P_1P_2} \)?
3. How do we represent vectors in three dimensions geometrically?
4. What is the dot product of two vectors, and how is it calculated?
5. How does the cross product of two vectors differ from the dot product?

**Tip:** When calculating vectors between points, always subtract the coordinates of the initial point from the terminal point to ensure the direction is correct.

Find the components of the vector P1P2 where P1(4,1,-3) and P2(4,5,-3).

Learn how to find the components of the vector between two points, P1(4,1,-3) and P2(4,5,-3), using the vector components formula. This example covers key concepts in 3D coordinate geometry and vectors, suitable for high school students in Grades 10-12.

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