A boy runs 7 blocks North, 9 blocks
Northeast, and 18 blocks West.
Determine the length of the displacement
vector that goes from the starting point to his
final position.
To find the length of the displacement vector, we need to calculate the magnitude of the final position vector.
First, we need to break down the boy's movements into their respective components. In this case, we can consider North as the positive y-direction and East as the positive x-direction.
Given:
North: 7 blocks
Northeast: 9 blocks
West: 18 blocks
We need to find the total displacement in the x and y directions separately, which we can then use to calculate the displacement vector.
In the y-direction:
Since the boy runs 7 blocks North, the displacement in the y-direction is 7 blocks North.
In the x-direction:
The boy runs 9 blocks Northeast, which can be broken down into its x and y components using the Pythagorean theorem. Since Northeast is a combination of North and East, we can say that the displacement is 9/sqrt(2) blocks in both the x and y directions.
Now, let's calculate the total displacement for the x and y directions:
In the y-direction: The displacement is 7 blocks North.
In the x-direction: The displacement is 9/sqrt(2) blocks Northeast, which is the same as 9/sqrt(2) blocks Southwest (since opposite directions cancel each other out).
To calculate the total displacement, we need to subtract the displacement in the x-direction since it is going in the opposite direction.
Total displacement in the x-direction: 0 - 9/sqrt(2) = -9/sqrt(2) blocks
Now, we can calculate the magnitude of the displacement vector:
Magnitude = sqrt((displacement in x)^2 + (displacement in y)^2)
Magnitude = sqrt((-9/sqrt(2))^2 + (7)^2)
Magnitude = sqrt(81/2 + 49)
Magnitude = sqrt(129.5)
Therefore, the length of the displacement vector is approximately 11.37 blocks.