A hiker makes four straight-line walks
A 31 km at 299 degrees
B 32 km at 210 degrees
C 29 km at 59 degrees
D 22 km at 150 degrees
in random directions and lengths starting at
position (41 km, 41 km).
Which vector will return the hiker to the
starting point? All angles are measured in a
counter-clockwise direction from the positive
x-axis.
Well, let's see if we can help this adventurous hiker find their way back home!
We'll start by breaking down the movements into their x and y components. The x-coordinate represents the horizontal movement, and the y-coordinate represents the vertical movement.
For the first walk A (31 km at 299 degrees), the x-component is 31 km * cos(299) and the y-component is 31 km * sin(299). With similar calculations, we find the components for the other walks as follows:
Walk A:
x-component: 31 km * cos(299) = -24.72 km
y-component: 31 km * sin(299) = 15.97 km
Walk B:
x-component: 32 km * cos(210) = -20.68 km
y-component: 32 km * sin(210) = -27.68 km
Walk C:
x-component: 29 km * cos(59) = 15.66 km
y-component: 29 km * sin(59) = 25.36 km
Walk D:
x-component: 22 km * cos(150) = -8.94 km
y-component: 22 km * sin(150) = 10.58 km
Now, let's sum up all the x and y components to determine the net displacement of the hiker:
Total x-component = -24.72 km - 20.68 km + 15.66 km - 8.94 km = -38.68 km
Total y-component = 15.97 km - 27.68 km + 25.36 km + 10.58 km = 24.23 km
So, after all those walks, the hiker's displacement from the starting point is approximately (-38.68 km, 24.23 km).
Now, to return to the starting point, the hiker needs to move the same distance but in the opposite direction. So, the vector that will bring them back to their starting point is approximately (38.68 km, -24.23 km).
Now, I hope your hiker buddy doesn't mind a little detour before getting back home! Happy journeying!
To find the vector that will return the hiker to the starting point, we need to consider the net displacement of the hiker after all four walks.
Let's break down each walk into their x and y components:
Walk A:
Distance = 31 km
Angle = 299 degrees
The x-component of walk A = 31 km * cos(299 degrees)
The y-component of walk A = 31 km * sin(299 degrees)
Walk B:
Distance = 32 km
Angle = 210 degrees
The x-component of walk B = 32 km * cos(210 degrees)
The y-component of walk B = 32 km * sin(210 degrees)
Walk C:
Distance = 29 km
Angle = 59 degrees
The x-component of walk C = 29 km * cos(59 degrees)
The y-component of walk C = 29 km * sin(59 degrees)
Walk D:
Distance = 22 km
Angle = 150 degrees
The x-component of walk D = 22 km * cos(150 degrees)
The y-component of walk D = 22 km * sin(150 degrees)
Now, let's calculate the net displacement by summing up the x and y components:
Net x-component = (x-component of A) + (x-component of B) + (x-component of C) + (x-component of D)
Net y-component = (y-component of A) + (y-component of B) + (y-component of C) + (y-component of D)
To find the overall vector, we use the Pythagorean theorem:
Magnitude of the vector = sqrt((Net x-component)^2 + (Net y-component)^2)
Now we can calculate the values step-by-step:
x-component of A = 31 km * cos(299 degrees) = -14.01 km
y-component of A = 31 km * sin(299 degrees) = 25.69 km
x-component of B = 32 km * cos(210 degrees) = -24.73 km
y-component of B = 32 km * sin(210 degrees) = -16.55 km
x-component of C = 29 km * cos(59 degrees) = 16.85 km
y-component of C = 29 km * sin(59 degrees) = 24.68 km
x-component of D = 22 km * cos(150 degrees) = -11 km
y-component of D = 22 km * sin(150 degrees) = 19.03 km
Net x-component = (-14.01 km) + (-24.73 km) + (16.85 km) + (-11 km) = -32.89 km
Net y-component = (25.69 km) + (-16.55 km) + (24.68 km) + (19.03 km) = 52.85 km
Magnitude of the vector = sqrt((-32.89 km)^2 + (52.85 km)^2) = 62.03 km
Therefore, the vector that will return the hiker to the starting point is approximately 62.03 km long.
To determine which vector will return the hiker to the starting point, we need to add up the individual vectors and see if their sum results in a vector that ends at the starting point.
We start with the position (41 km, 41 km), and then add up the four vectors based on their length and direction.
First, let's convert the given angles to standard position angles (measured counterclockwise from the positive x-axis). The angles are already in counterclockwise direction, so we don't need to make any changes.
Now, let's break down the given vectors into their x and y components using trigonometry:
Vector A: magnitude 31 km, angle 299 degrees
Ax = 31 km * cos(299) (x-component)
Ay = 31 km * sin(299) (y-component)
Vector B: magnitude 32 km, angle 210 degrees
Bx = 32 km * cos(210) (x-component)
By = 32 km * sin(210) (y-component)
Vector C: magnitude 29 km, angle 59 degrees
Cx = 29 km * cos(59) (x-component)
Cy = 29 km * sin(59) (y-component)
Vector D: magnitude 22 km, angle 150 degrees
Dx = 22 km * cos(150) (x-component)
Dy = 22 km * sin(150) (y-component)
Now, let's calculate the x and y components for each vector:
Ax = 31 km * cos(299) ≈ 24.61 km
Ay = 31 km * sin(299) ≈ -12.48 km
Bx = 32 km * cos(210) ≈ -19.87 km
By = 32 km * sin(210) ≈ -27.71 km
Cx = 29 km * cos(59) ≈ 15.08 km
Cy = 29 km * sin(59) ≈ 24.77 km
Dx = 22 km * cos(150) ≈ -11 km
Dy = 22 km * sin(150) ≈ -19 km
Now, let's add up the x and y components of all the vectors:
Total x-component = Ax + Bx + Cx + Dx ≈ 24.61 km + (-19.87 km) + 15.08 km + (-11 km) ≈ 8.82 km
Total y-component = Ay + By + Cy + Dy ≈ (-12.48 km) + (-27.71 km) + 24.77 km + (-19 km) ≈ -34.42 km
Therefore, the vector that returns the hiker to the starting point has an x-component of approximately 8.82 km and a y-component of approximately -34.42 km.
To represent this vector, we can use the form (x, y): (8.82 km, -34.42 km).