A: A boy runs 7.2 blocks North, 12.2 blocks Northeast, and 12.3 blocks West.
Determine the length of the displacement vector that goes from the starting point to his final position.
B: Determine the direction of the displacement vector. Use counterclockwise as the positive angular direction, between the limits of −180◦ and +180◦ measured from East.
Answer in units ◦
North = 7.2 + 12.2 cos 45
West = 12.3 - 12.2 cos 45
answer = sqrt [ North^2 + West^2 ] blocks
answer is in northwest quadrant
tan angle = North/West, find angle which is above -x axis
subtract angle from 180
To find the length of the displacement vector, we can simply calculate the Pythagorean sum of the three distances traveled.
First, we square each distance traveled:
(7.2)^2 = 51.84
(12.2)^2 = 148.84
(12.3)^2 = 151.29
Then, we add up these squared distances:
51.84 + 148.84 + 151.29 = 351.97
Finally, we take the square root of the sum to find the magnitude of the displacement vector:
√(351.97) = 18.76 blocks (rounded to two decimal places)
Therefore, the length of the displacement vector from the starting point to the final position is approximately 18.76 blocks.
To determine the direction of the displacement vector, we can use trigonometry.
First, we can use the distances traveled in the North and West directions to find the angle made with the East direction. In this case, the angle is in the third quadrant, so we need to calculate the angle from the positive x-axis and count it as negative:
θ = atan(North Distance / West Distance)
= atan(7.2 / -12.3)
≈ -30.54 degrees
Since the angle in the question is measured counterclockwise from the positive x-axis, we can convert this angle to the desired range of -180 degrees to +180 degrees by subtracting 360 degrees if the angle is greater than 180 degrees or adding 360 degrees if the angle is less than -180 degrees:
If the angle is greater than 180 degrees:
θ -= 360
θ = -30.54 - 360
θ ≈ -390.54 degrees
If the angle is less than -180 degrees:
θ += 360
θ = -30.54 + 360
θ ≈ 329.46 degrees
Therefore, the direction of the displacement vector from the starting point to the final position is approximately -30.54 degrees (or 329.46 degrees when converted to the desired range of -180 to +180 degrees).