A boy runs 3 blocks north, 4 blocks north east, and 5 blocks west. Determine the length and the direction displacement vector that goes from the starting point to his final position
To determine the length and direction of the displacement vector, we need to find the resulting displacement from the starting point to the final position.
Considering the North direction as the positive y-axis and the East direction as the positive x-axis, let's break down the given displacements into their x and y components:
- The displacement of running 3 blocks north can be represented as (0, 3) since there is no change in the x-coordinate and a positive change of 3 in the y-coordinate.
- The displacement of running 4 blocks northeast can be represented as (4cos(45°), 4sin(45°)), where the angle 45° is formed by the northeast direction with the positive x-axis. Simplifying, (4√2/2, 4√2/2) or approximately (2.83, 2.83).
- The displacement of running 5 blocks west can be represented as (-5, 0) since there is a negative change of 5 in the x-coordinate and no change in the y-coordinate.
Now, let's add these displacements together to find the resultant displacement:
Resultant displacement = (0 + 2.83 - 5, 3 + 2.83 + 0)
= (-2.17, 5.83)
The length of the displacement vector can be found using the Pythagorean theorem:
Length = √((-2.17)^2 + (5.83)^2)
= √(4.7089 + 34.0489)
= √38.7578
≈ 6.23
Therefore, the length of the displacement vector is approximately 6.23 blocks.
The direction of the displacement vector can be found using trigonometry:
Direction = arctan(5.83 / -2.17)
≈ arctan(-2.689)
≈ -69.94°
Note: The negative angle indicates that the direction is towards the fourth quadrant (southwest) with respect to the positive x-axis.
Hence, the length of the displacement vector is approximately 6.23 blocks, and the direction is approximately -69.94° (southwest).
To determine the length and direction of the displacement vector, we can use the Pythagorean theorem and trigonometric ratios.
First, let's break down the boy's movements into their vertical and horizontal components.
He runs 3 blocks north, which means he moves 3 blocks in the positive y-direction (upwards).
Next, he runs 4 blocks northeast. We can split this movement into its vertical and horizontal components using trigonometry. In a right triangle, the angle between the diagonal side (northeast) and the horizontal side (east) is 45 degrees. Using this angle, we can find the vertical and horizontal components.
The vertical component is obtained by multiplying the hypotenuse (4 blocks) by the sine of the angle (45 degrees), so the boy moves 4 * sin(45) blocks in the positive y-direction.
The horizontal component is obtained by multiplying the hypotenuse (4 blocks) by the cosine of the angle (45 degrees), so the boy moves 4 * cos(45) blocks in the positive x-direction (east).
Lastly, he runs 5 blocks west, so he moves 5 blocks in the negative x-direction (west).
Now let's calculate the total displacement:
In the vertical direction, the boy moves 3 blocks north initially, then an additional 4 * sin(45) blocks north. So the total vertical displacement is 3 + 4 * sin(45).
In the horizontal direction, the boy moves 4 * cos(45) blocks east initially, then 5 blocks west. The total horizontal displacement is 4 * cos(45) - 5.
We can now use the Pythagorean theorem to find the length of the displacement vector. The displacement vector is the hypotenuse of a right triangle, where the vertical and horizontal displacements are the other two sides. The length of the displacement vector is given by:
length = sqrt(vertical displacement^2 + horizontal displacement^2)
= sqrt((3 + 4 * sin(45))^2 + (4 * cos(45) - 5)^2)
To find the direction of the displacement vector, we need to calculate the angle it makes with the positive x-axis. We can use the inverse tangent function (arctan) to find this angle:
angle = arctan(vertical displacement / horizontal displacement)
= arctan((3 + 4 * sin(45)) / (4 * cos(45) - 5))
Therefore, the length of the displacement vector is sqrt((3 + 4 * sin(45))^2 + (4 * cos(45) - 5)^2), and the direction of the displacement vector is arctan((3 + 4 * sin(45)) / (4 * cos(45) - 5)).