vector A has a magnitude of 57 m and points in a direction 20 degrees below the positive x axis. vector B has magnitude of 70 m and points in a direction 57 degrees above the positive x axis. Find the magnitude of the vector D = A-B.
A = 57m @ -20 deg. B = 70m @ 57 deg.
-20 = 340 deg.(ccw from 0 deg.),
x =hor= 57*Cos(340) + 70*Cos(57)=91.7m
y =ver= 57*Sin(340) + 70*Sin(57)=39.2m
D^2 = X^2 + Y^2 =(91.7)^2+(39.2)^2=
D^2 = 8408.89 + 1536.64 = 9945.53,
D = sqrt(9945.53) = 99.7 Meters.
To find the magnitude of vector D = A - B, we need to subtract the components of vector B from the components of vector A and then find the magnitude of the resulting vector.
Let's break down the given information and calculate the components of A and B.
For vector A:
- Magnitude: 57 m
- Direction: 20 degrees below the positive x-axis
To find the x-component of A, we can use the cosine function:
A_x = A_magnitude * cos(angle)
A_x = 57 m * cos(20 degrees)
To find the y-component of A, we can use the sine function:
A_y = A_magnitude * sin(angle)
A_y = 57 m * sin(20 degrees)
Similarly, for vector B:
- Magnitude: 70 m
- Direction: 57 degrees above the positive x-axis
To find the x-component of B:
B_x = B_magnitude * cos(angle)
B_x = 70 m * cos(57 degrees)
To find the y-component of B:
B_y = B_magnitude * sin(angle)
B_y = 70 m * sin(57 degrees)
Next, subtract the components of B from the components of A to get the components of vector D:
D_x = A_x - B_x
D_y = A_y - B_y
Finally, find the magnitude of vector D using the Pythagorean theorem:
D_magnitude = sqrt(D_x^2 + D_y^2)
Let's calculate it step by step.
A_x = 57 m * cos(20 degrees) ≈ 53.95 m
A_y = 57 m * sin(20 degrees) ≈ 19.37 m
B_x = 70 m * cos(57 degrees) ≈ 36.11 m
B_y = 70 m * sin(57 degrees) ≈ 56.42 m
D_x = A_x - B_x ≈ 53.95 m - 36.11 m ≈ 17.84 m
D_y = A_y - B_y ≈ 19.37 m - 56.42 m ≈ -37.05 m
D_magnitude = sqrt(D_x^2 + D_y^2)
D_magnitude = sqrt((17.84 m)^2 + (-37.05 m)^2) ≈ sqrt(318.97 m^2 + 1371.03 m^2) ≈ sqrt(1690 m^2) ≈ 41.1 m
Therefore, the magnitude of vector D = A - B is approximately 41.1 m.
Remember to round your final answer to an appropriate number of significant figures based on the available information.