https://file.aiquickdraw.com/mathbot/file/1726029019035gyl9y31z.jpg

The equation in the image shows the vector relationship:

\[
\overrightarrow{AP} = k \overrightarrow{AB} \quad (k \neq 0)
\]

This indicates that the vector \(\overrightarrow{AP}\) is a scalar multiple of the vector \(\overrightarrow{AB}\), with \(k\) being the scalar. The condition \(k \neq 0\) ensures that \(P\) is not the same as point \(A\) (i.e., there is a real displacement from \(A\) towards \(P\)).

Key interpretations:
1. The points \(A\), \(B\), and \(P\) are collinear since one vector is a scalar multiple of the other, meaning the two vectors point in the same or opposite directions.
2. The scalar \(k\) defines how far along or beyond point \(B\) the point \(P\) lies relative to \(A\). If \(k > 0\), \(P\) lies on the line segment extending in the direction from \(A\) to \(B\). If \(k < 0\), \(P\) lies on the line extending in the opposite direction beyond \(A\).
3. When \(k = 1\), \(P\) coincides with \(B\), and when \(0 < k < 1\), \(P\) is between \(A\) and \(B\).

Would you like further clarification or more detailed calculations based on this vector equation?

Here are 5 related questions to explore:
1. What is the geometric interpretation of different values of \(k\) in the context of points \(A\), \(B\), and \(P\)?
2. How do you find the coordinates of \(P\) if the coordinates of \(A\) and \(B\) are known?
3. What happens if \(k = -1\)? Where does point \(P\) lie relative to points \(A\) and \(B\)?
4. Can you express the position of point \(P\) in terms of parametric equations?
5. How would this concept change in three-dimensional space?

**Tip**: The parameter \(k\) provides a powerful way to represent any point on the line determined by two points \(A\) and \(B\) using vectors.

What is the meaning of the vector equation \( \overrightarrow{AP} = k \overrightarrow{AB} \) where \( k \neq 0 \)?

Understand the vector equation \( \overrightarrow{AP} = k \overrightarrow{AB} \) where k ≠ 0, representing the relationship between vectors AP and AB through scalar multiplication. Learn about collinearity of points A, B, and P, and how k affects the position of P relative to A and B.

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