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.