Ricardo and Jane are standing under a tree in the middle of a pasture. An argument ensues, and they walk away in different directions. Ricardo walks 22.0 in a direction 60.0 west of north. Jane walks 10.0 in a direction 30.0 south of west. They then stop and turn to face each other.
Your question is incomplete. You described two displacement vectors but never asked a question about them. You also did not provide dimensions for the displacement vectors. (feet, meters, km?)
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To find their final positions and the distance between them, we can break down their movements into components and use vector addition.
First, let's find the components of Ricardo's and Jane's initial movements:
Ricardo walks 22.0 in a direction 60.0 west of north. Let's break this down into north and west components. The magnitude of the north component (Ricardo's displacement in the north direction) can be found using trigonometry:
north component = magnitude * cos(angle)
north component = 22.0 * cos(60°)
north component = 22.0 * 0.5
north component = 11.0
The magnitude of the west component (Ricardo's displacement in the west direction) can be found using trigonometry:
west component = magnitude * sin(angle)
west component = 22.0 * sin(60°)
west component = 22.0 * √3 / 2
west component ≈ 19.0
So Ricardo's initial position relative to the tree is 11.0 units north and 19.0 units west.
Jane walks 10.0 in a direction 30.0 south of west. Let's break this down into south and west components. The magnitude of the south component (Jane's displacement in the south direction) can be found using trigonometry:
south component = magnitude * sin(angle)
south component = 10.0 * sin(30°)
south component = 10.0 * 0.5
south component = 5.0
The magnitude of the west component (Jane's displacement in the west direction) can be found using trigonometry:
west component = magnitude * cos(angle)
west component = 10.0 * cos(30°)
west component = 10.0 * √3 / 2
west component ≈ 8.7
So Jane's initial position relative to the tree is 5.0 units south and 8.7 units west.
Now, let's find their final positions:
After Ricardo and Jane walk in opposite directions, their final positions will be the sums of their initial positions and their respective movements. Ricardo's final position is obtained by adding his initial position to his movement:
Ricardo's final position = Ricardo's initial position + Ricardo's movement
Ricardo's final position = (11.0 north, 19.0 west) + (11.0 north, -19.0 east)
Ricardo's final position = (11.0 north, 0.0 west)
Jane's final position is obtained by adding her initial position to her movement:
Jane's final position = Jane's initial position + Jane's movement
Jane's final position = (5.0 south, 8.7 west) + (-5.0 south, 8.7 west)
Jane's final position = (0.0 south, 17.4 west)
Now that we have their final positions, we can find the distance between them by considering them as vectors:
Distance between Ricardo and Jane = √((Ricardo's final position - Jane's final position)^2)
Distance between Ricardo and Jane = √((11.0 - 0.0)^2 + (0.0 - 17.4)^2)
Distance between Ricardo and Jane = √(121.0 + 302.76)
Distance between Ricardo and Jane ≈ √423.76
Distance between Ricardo and Jane ≈ 20.6
Therefore, Ricardo and Jane are approximately 20.6 units away from each other after they stop and turn to face each other.