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Displacement components help us understand how far and in which direction an object has moved from its starting point. With vector analysis, we break down movements into north-south and east-west components.
- **Ricardo's Movement:** He travels 26.0 m at 60.0° west of north. To determine the exact distance moved in the north and west directions, we use trigonometric functions: - Northward: 26.0 m \( \cos(60°) \) = 13.0 m - Westward: 26.0 m \( \sin(60°) \) = 22.5 m - **Jane's Movement:** She walks 16.0 m at 30.0° south of west. Her displacement components are: - Westward: 16.0 m \( \cos(30°) \) = 13.9 m - Southward: 16.0 m \( \sin(30°) \) = 8.0 m By splitting their paths, we simplify the analysis into straightforward right-angle movements. This makes further calculations, like determining distances and directions, much more manageable.

The Pythagorean theorem is vital in vector analysis for calculating the distance between two points based on perpendicular components. It states that in a right triangle, the square of the hypotenuse (longest side) is equal to the sum of the squares of the other two sides. For Ricardo and Jane, their combined north-south and east-west displacements form a right triangle, with:- North-South: 5.0 m - East-West: 36.4 m To find the distance between them, imagine these as two sides forming the right angle, and use:\[ \text{Distance} = \sqrt{(5.0 \text{ m})^2 + (36.4 \text{ m})^2} \]This equation allows us to compute:\[ \sqrt{25.0 + 1324.96} = \sqrt{1349.96} \approx 36.7 \text{ m} \]This result means Ricardo and Jane are roughly 36.7 meters apart. Such calculations are essential in physics and mathematics to avoid measurement errors.

Knowing the direction angle is crucial for understanding which way to travel. Here, we calculate the angle at which Ricardo must walk to head directly towards Jane. This angle is determined by comparing their east-west and north-south displacements.To calculate the angle, use the arctangent (\( \arctan \)) function:\[ \text{Angle} = \arctan\left(\frac{36.4 \text{ (m)}}{5.0 \text{ (m)}}\right) \approx 82.0° \]This calculation gives us a precise angle of 82.0°. However, this angle isn’t just any angle; it’s measured west of north. This means that if Ricardo wants to walk directly towards Jane, he should adjust his path to 82.0° to the west from the north line. Understanding such angle calculations can help in navigation and determining the correct direction on maps or in other vector-based activities.