Multiple Concept Example 9 provides background pertinent to this problem. The magnitudes of the four displacement vectors shown in the drawing are A = 17.0 m, B = 10.0 m, C = 11.0 m, and D = 23.0 m. Determine the (a) magnitude and (b) direction for the resultant that occurs when these vectors are added together. Specify the direction as a positive (counterclockwise) angle from the +x axis.
A: direction is west of north with a degree of 20.0
B: direction is north of center no degree
C: direction is south of west with a degree of 35.0
D: direction is south of east with a degree of 50.0
havent I answered this? Convert the angles measured from x axis counter clockwise. assuming your East direction is x axis..
a. angle is 20+90
b. angle is 90
c. 180+35
d. 360-50
To find the resultant magnitude and direction when the four displacement vectors are added together, we will use vector addition. We'll start by representing the given vectors in Cartesian coordinate form.
Let's assign positive x-axis towards the east and positive y-axis towards the north. Based on the given information, we can represent the vectors as follows:
A = 17.0 m, pointing west (negative x-axis)
B = 10.0 m, pointing north (positive y-axis)
C = 11.0 m, pointing south of west (negative x-axis and negative y-axis)
D = 23.0 m, pointing south of east (positive x-axis and negative y-axis)
Now, let's break down the vectors into their x and y components:
For vector A:
Ax = -17.0 m (negative because it points towards the negative x-axis)
Ay = 0 m (no y-component as it only points in the x-direction)
For vector B:
Bx = 0 m (no x-component as it only points in the y-direction)
By = 10.0 m (positive because it points towards the positive y-axis)
For vector C:
Cx = -11.0 m (negative because it points towards the negative x-axis)
Cy = -11.0 m (negative because it points towards the negative y-axis)
For vector D:
Dx = 23.0 m (positive because it points towards the positive x-axis)
Dy = -23.0 m (negative because it points towards the negative y-axis)
Next, we'll add the x and y components separately to find the resultant x-component (Rx) and resultant y-component (Ry):
Rx = Ax + Bx + Cx + Dx
Ry = Ay + By + Cy + Dy
Substituting the values we have:
Rx = -17.0 m + 0 m - 11.0 m + 23.0 m = -5.0 m
Ry = 0 m + 10.0 m - 11.0 m - 23.0 m = -24.0 m
Now, we can find the magnitude (R) of the resultant vector using the Pythagorean theorem:
R = sqrt(Rx^2 + Ry^2)
Substituting the values we have:
R = sqrt((-5.0 m)^2 + (-24.0 m)^2)
R ≈ 24.83 m
So, the magnitude of the resultant vector is approximately 24.83 m.
To determine the direction of the resultant vector, we can use the inverse tangent function. The direction is given as a positive (counterclockwise) angle from the +x axis.
θ = tan^(-1)(Ry/Rx)
Substituting the values we have:
θ = tan^(-1)(-24.0 m / -5.0 m)
θ ≈ 78.69 degrees
Since the question mentions that the direction is "west of north," which means the angle is measured counterclockwise from the +x axis, we can find the angle by subtracting θ from 90 degrees:
90 degrees - 78.69 degrees ≈ 11.31 degrees
Therefore, the direction of the resultant vector is approximately 11.31 degrees counterclockwise from the +x axis.
Based on the given options, none of the choices match the calculated magnitude and direction.