Understanding Joanna's Displacement in Geometry

The problem of finding Joanna's displacement involves a series of vector movements in a two-dimensional plane. Let's break down her journey step by step and understand how to calculate her final displacement.

Visualizing Joanna's Movements

Joanna starts at the origin point (0, 0) and follows these steps:

Travels 5 miles north, so her position is (0, 5). Travels 3 miles west, so her position is (-3, 5). Travels 2 miles south, so her position is (-3, 3).

Now, to find her displacement, we need to determine the straight-line distance from her starting point (0, 0) to her final position (-3, 3).

Calculating Displacement Using the Distance Formula

The distance d between two points x1, y1 and x2, y2 is given by the formula:

[d sqrt{(x_2 - x_1)^2 (y_2 - y_1)^2}]

Substituting Joanna's Coordinates

Starting point: (0, 0) Final point: (-3, 3)

[d sqrt{(-3 - 0)^2 (3 - 0)^2}]

Simplifying this expression:

[d sqrt{(-3)^2 3^2}] [d sqrt{9 9}] [d sqrt{18}] [d 3sqrt{2} text{miles} approx 4.24 text{miles}]

Therefore, Joanna's displacement is approximately 4.24 miles in a direction west of north.

Geometric Interpretation of Joanna's Displacement

Joanna's displacement can be visualized as a vector that points from her starting point to her final position. In a coordinate plane, this vector can be represented as:

[text{Displacement vector} 5mathbf{j} - 3mathbf{i} - 2mathbf{j} -3mathbf{i} - 3mathbf{j}]

The magnitude of the displacement vector is given by:

[|text{Displacement vector}| sqrt{(-3)^2 (-3)^2} 3sqrt{2} approx 4.24 text{miles}]

The direction of the displacement vector can be found using the angle from the positive x-axis (east) to the direction of the vector. This angle is:

[theta arctanleft(frac{-3}{-3}right) 45°]

So, Joanna's displacement is approximately 4.24 miles in the direction 45° west of north.

Special Cases in Joanna's Displacement

The problem of calculating displacement can have special cases depending on the starting point. For example:

If Joanna starts at a point that is 5.477 miles south of the North Pole, her first leg of travel will take her to a point 0.477 miles south of the North Pole. The second leg will return her to the same point. The third leg will trace 2 miles back on the first leg, making her final displacement 3 miles north. If a larger displacement is required, Joanna could start at a point that is 5.955 miles south of the North Pole, which would make her final displacement even more significant.

Understanding vector movements in these special cases helps in comprehending the true nature of displacement in a three-dimensional context.