⦁ Find the resultant vector (magnitude and direction) for the sum of:

vector A which has magnitude 365 N at 48° south of west,
vector B which has magnitude 224 N at 26° south of east, and
vector C which has magnitude 488 N at 12° west of north.
Express the direction of the resultant relative to due west.

F = 365N[228o]CCW+224N[334o]CCW+488N [102o]CCW.

Fx=365*Cos228 + 224*Cos334 + 488*cos102
= -144.4 N.

Fy=365*sin228 + 224*sin334 + 488*sin102
= 107.9 N.

Tan Ar = Fy/Fx = 107.9/-144.4 = -0.74718
Ar = -36.8o = Reference angle.
A = -36.8 + 180 = 143.2o CCW = 36.8o N.
of W.

F = Fx/Cos A = -144.4/Cos143.2 = 180.3 N
[143.2]CCW = 180.3N[36.8o] North of West.

fire

To find the resultant vector, we need to add the three given vectors A, B, and C.

First, let's break down each vector into its x and y components using trigonometry.

For vector A:
- Magnitude: 365 N
- Direction: 48° south of west

To find the x-component of A, we use cosine:
A_x = A_magnitude * cos(A_direction)
= 365 N * cos(48°)

To find the y-component of A, we use sine:
A_y = A_magnitude * sin(A_direction)
= 365 N * sin(48°)

Similarly, we can break down vectors B and C into their x and y components:

For vector B:
- Magnitude: 224 N
- Direction: 26° south of east

B_x = B_magnitude * cos(B_direction)
= 224 N * cos(26°)

B_y = B_magnitude * sin(B_direction)
= 224 N * sin(26°)

For vector C:
- Magnitude: 488 N
- Direction: 12° west of north

C_x = C_magnitude * cos(C_direction)
= 488 N * cos(12°)

C_y = C_magnitude * sin(C_direction)
= 488 N * sin(12°)

Now, we can calculate the x and y components of the resultant vector R by adding the corresponding components of A, B, and C:

R_x = A_x + B_x + C_x
= (365 N * cos(48°)) + (224 N * cos(26°)) + (488 N * cos(12°))

R_y = A_y + B_y + C_y
= (365 N * sin(48°)) + (224 N * sin(26°)) + (488 N * sin(12°))

To find the magnitude of the resultant vector R, we use the Pythagorean theorem:

R_magnitude = sqrt(R_x^2 + R_y^2)

Finally, to find the direction of the resultant vector R relative to due west, we can use the inverse tangent (arctan) function:

R_direction = arctan(R_y / R_x)

Now, you can substitute the values of A_x, A_y, B_x, B_y, C_x, C_y, R_x, and R_y into the above equations to calculate the magnitude and direction of the resultant vector R.