Multiple-Concept Example 9 provides background pertinent to this problem. The magnitudes of the four displacement vectors shown in the

Question

drawing are A = 16,0 m, B = 11.0 m, C = 12.0 m, and D = 26.0 m

Determine the magnitude and directional angle for the resultant tha occurs when these vectors are added together

Answer 1

To find the magnitude and directional angle of the resultant vector, we can use the Pythagorean theorem and the tangent function.

First, let's add the vectors together:
Resultant vector = A + B + C + D

Resultant magnitude:
Magnitude = sqrt(A^2 + B^2 + C^2 + D^2)
Magnitude = sqrt((16.0)^2 + (11.0)^2 + (12.0)^2 + (26.0)^2)
Magnitude ≈ sqrt(256.0 + 121.0 + 144.0 + 676.0)
Magnitude ≈ sqrt(1197.0)
Magnitude ≈ 34.6 m

To find the directional angle, we need to find the components of the resultant vector in the x-axis and y-axis.

Component in the x-axis:
Rx = Ax + Bx + Cx + Dx
Rx = 16.0 m + 11.0 m + 12.0 m + 26.0 m
Rx = 65.0 m

Component in the y-axis:
Ry = Ay + By + Cy + Dy
Ry = 0 m + 0 m + 0 m + 0 m
Ry = 0 m

Now, we can use the tangent function to find the directional angle:

Directional angle = tan^(-1)(Ry / Rx)
Directional angle = tan^(-1)(0 / 65.0)
Directional angle ≈ tan^(-1)(0)
Directional angle ≈ 0°

Therefore, the magnitude of the resultant vector is approximately 34.6 m and the directional angle is 0° (or along the positive x-axis).