a delivery truck travels 9 blocks north, 7 blocks west and 12 blocks southwest. What is the final displacement from the origin? blocks are equal length

do 12 southwest first

distance north = -12 cos 45
distance west = 12 sin 45

now total north
distance north = 9 -12 cos 45 = .515
now total west
distance west = 7 + 12 sin 45 = 15.5

To find the final displacement from the origin, we need to consider the overall movement in terms of distance and direction.

First, let's break down the given movements:
- The truck travels 9 blocks north (which means going up on the coordinate plane).
- Then, it goes 7 blocks west (which means going left on the coordinate plane).
- Finally, it moves 12 blocks southwest (which means going diagonally downwards and to the left).

Now, let's visualize this:

| |
| |
| X | <-- Final position
| |
| |
|-------|-------|-------|-------|-------|

Initial position (Origin)

To determine the final displacement from the origin, we can use vector addition. We will take the sum of the x and y components of the individual movements.

For the north movement:
- The truck moves 9 blocks north, so the y-component is 9 and there is no change in the x-component (since it moves purely in the y direction).

For the west movement:
- The truck moves 7 blocks west, so the x-component becomes -7 (negative since it's in the opposite direction) and there's no change in the y-component.

For the southwest movement:
- The truck moves 12 blocks southwest, so we need to break it down into x and y components. Since it's diagonal, the x and y components have the same magnitude. Using the Pythagorean theorem, the magnitude can be found:
magnitude = sqrt(12^2 + 12^2) = sqrt(144 + 144) = sqrt(288) = 12√2
- The southwest movement has an angle of 225 degrees. We can now determine the x and y components using trigonometry:
x-component = magnitude * cos(225°) ≈ 12√2 * (-1/√2) = -12
y-component = magnitude * sin(225°) ≈ 12√2 * (-1/√2) = -12

Now, let's calculate the final displacement:
- x-component = (-7) + (-12) = -19
- y-component = 9 + (-12) = -3

Therefore, the final displacement from the origin is 19 blocks west and 3 blocks south, or simply (-19, -3).