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In physics, vectors are essential tools because they have both magnitude (size) and direction. When a problem involves multiple vectors, like our shopper's path, we need to use vector addition to determine the overall effect.

- **Vectors**: Think of vectors as arrows, each with a certain length and pointing in a specific direction.
- **Vector Addition**: When adding vectors, you place the tail of the next vector at the head of the previous one. This can help in visualizing and calculating the resultant vector — the vector that represents the total displacement.

In our shopper's journey, the total displacement can be seen as adding three separate vectors: south (40.0 m), west (15.0 m), and north (20.0 m). It's important to keep track of directions because they impact how vectors add up. For instance, the opposite directions (north and south) partially cancel each other out.

Rectilinear motion refers to movement along a straight line. In many physics problems, it can be simplified to horizontal or vertical components.

- **Straight Paths**: Even though the shopper turns, each segment of the path is a straight line.
- **Component Breakdown**: By breaking movement down into individual segments (south, west, north), we can analyze each one separately.

This simplifies the process of finding displacement because we can focus on one direction at a time. For instance, the movement south and north are in opposite directions along the same straight line, allowing us to calculate the resulting displacement directly by subtraction.

Determining the magnitude and direction of the resultant displacement is the key objective in problems involving vector operations.

- **Magnitude**: This refers to the length of the resultant vector and is often calculated using the Pythagorean theorem when dealing with perpendicular components. For our shopper, we can combine the south and west displacements to find the overall magnitude of his displacement.
- **Direction**: The direction of the resultant vector is given relative to a reference direction. For instance, a vector pointing southwest indicates a shift both towards the south and the west.

By understanding both the magnitude and direction, one can fully describe the resultant displacement of the shopper, as being 20.0 m south and 15.0 m west. This combination results in a diagonal path toward the southwest.