A bug starts at point A, crawls 8.0cm east, then 5.0cm south,3.0cm west,and 4.0cm north to point B. (a)How far north and east is B from A? (b) Find the displacement from A to B both graphically and algebraically.

To solve the problem of the bug's journey from point A to point B, we can break it down into two parts: determining the final position of point B relative to point A, and then calculating the displacement both graphically and algebraically. Let's tackle each part step by step.

Determining Position B Relative to A

First, we need to track the bug's movements in terms of coordinates. We can start by assuming point A is at the origin (0, 0) on a coordinate plane.

• The bug crawls 8.0 cm east: This moves it to (8.0, 0).
• Next, it crawls 5.0 cm south: This changes its position to (8.0, -5.0).
• Then, it crawls 3.0 cm west: This moves it to (5.0, -5.0).
• Finally, it crawls 4.0 cm north: This results in the final position at (5.0, -1.0).

Coordinates of Point B

After all these movements, point B is located at the coordinates (5.0, -1.0). Now, we can determine how far north and east point B is from point A.

• **East-West Position**: The x-coordinate of point B is 5.0 cm, indicating it is 5.0 cm east of point A.
• **North-South Position**: The y-coordinate of point B is -1.0 cm, which means it is 1.0 cm south of point A.

Calculating Displacement

Displacement is defined as the shortest distance from the initial position to the final position, along with the direction. We can find this both graphically and algebraically.

Graphical Approach

To visualize the displacement, you can draw a right triangle where:

• The horizontal leg represents the east-west movement (5.0 cm east).
• The vertical leg represents the north-south movement (1.0 cm south).

Using the Pythagorean theorem, the displacement (d) can be calculated as follows:

d = √(east² + south²) = √(5.0² + 1.0²) = √(25 + 1) = √26 ≈ 5.1 cm.

Algebraic Approach

Algebraically, we can also express the displacement vector from A to B as:

Displacement vector = (x₂ - x₁, y₂ - y₁) = (5.0 - 0, -1.0 - 0) = (5.0, -1.0).

To find the magnitude of this vector, we again use the Pythagorean theorem:

Magnitude = √(5.0² + (-1.0)²) = √(25 + 1) = √26 ≈ 5.1 cm.

Summary of Results

In summary:

• Point B is located 5.0 cm east and 1.0 cm south of point A.
• The displacement from A to B is approximately 5.1 cm in a direction that can be described as east-southeast.

This methodical approach helps us understand both the position and the displacement of the bug's journey effectively. If you have any further questions or need clarification on any part, feel free to ask!