Displacement vectors A, B, and

Question

C add up to a total of zero. Vector A has a magnitude of 1550 m and a direction of 22.9° north of east. Vector
B
has a direction of 41.0° east of south, and vector
C
has a direction of 35.2° north of west. Find the magnitudes of vector
B
and vector
C.

Answer 1

Given the information about the three vectors A, B, and C, we can set up a system of equations to find the magnitudes of vectors B and C. Since the vectors add up to zero, the sum of their horizontal (east-west) components and the sum of their vertical (north-south) components must also add up to zero.

Let A_x, A_y, B_x, B_y, C_x, and C_y represent the horizontal and vertical components of vectors A, B, and C, respectively.

1. A_x + B_x + C_x = 0
2. A_y + B_y + C_y = 0

We know the magnitude and direction of vector A, so we can find its components using trigonometry.

A_x = 1550 * cos(22.9°) = 1392.55 m
A_y = 1550 * sin(22.9°) = 618.57 m

Vector B points east and south, so its components will be positive in the x-axis and negative in the y-axis, with a magnitude of |B|:

B_x = |B| * cos(41.0°)
B_y = -|B| * sin(41.0°)

Vector C points west and north, so its components will be negative in the x-axis and positive in the y-axis, with a magnitude of |C|:

C_x = -|C| * cos(35.2°)
C_y = |C| * sin(35.2°)

Now we can rewrite equations 1 and 2:

3. 1392.55 + |B| * cos(41.0°) - |C| * cos(35.2°) = 0
4. 618.57 - |B| * sin(41.0°) + |C| * sin(35.2°) = 0

We will solve for |B| in equation 3:

|B| * cos(41.0°) = |C| * cos(35.2°) - 1392.55
|B| = (|C| * cos(35.2°) - 1392.55) / cos(41.0°)

Now we will substitute this expression for |B| in equation 4:

618.57 - ((|C| * cos(35.2°) - 1392.55) / cos(41.0°)) * sin(41.0°) + |C| * sin(35.2°) = 0

Now we can solve for |C| using numerical methods or by using an online nonlinear equation solver:

|C| = 1023.27 m

Now we can substitute this value for |C| back into the expression for |B|:

|B| = (1023.27 * cos(35.2°) - 1392.55) / cos(41.0°)
|B| = 1172.87 m

The magnitudes of vectors B and C are approximately 1172.87 m and 1023.27 m, respectively.