Consider two vectors F~

1 with magnitude 44 N
inclined at 163

and F~
2 with magnitude 72 N
inclined at 236

, measured from the positive
x axis with counterclockwise positive.
What is the magnitude of the resultant vector F~
1 + F~
2 ? Draw the vectors to scale on a
graph to determine the answer.
1. 130 N
2. 98 N
3. 123 N
4. 95 N
5. 52 N
6. 72 N

What is the direction of this resultant vector (between the limits of −180◦
and +180◦
from the positive x-axis) ?
1. 103◦
2. 6◦
3. 23◦
4. 37◦
5. −38◦
6. −150◦

To find the magnitude of the resultant vector F1 + F2, you can use the parallelogram rule of vector addition. The parallelogram rule states that if two vectors are added together, the resultant vector is the diagonal of the parallelogram formed by the two vectors.

To draw the vectors to scale on a graph, you first need to determine the x and y components of each vector. The x component can be found by multiplying the magnitude of the vector by the cosine of the angle, and the y component can be found by multiplying the magnitude of the vector by the sine of the angle.

For vector F1 with magnitude 44 N inclined at 163°:
F1x = 44 N * cos(163°)
F1y = 44 N * sin(163°)

For vector F2 with magnitude 72 N inclined at 236°:
F2x = 72 N * cos(236°)
F2y = 72 N * sin(236°)

Next, you can plot these components on a graph. The x components will give you horizontal displacements, and the y components will give you vertical displacements. Start by plotting the x components of each vector, with the tail of each vector at the origin.

From the graph, you can see that the horizontal displacement is the sum of the x components, and the vertical displacement is the sum of the y components. The resultant vector is the vector that connects the tail of the first vector to the head of the second vector.

Finally, you can use the Pythagorean theorem to find the magnitude of the resultant vector:
Magnitude of resultant vector = sqrt((horizontal displacement)^2 + (vertical displacement)^2)

For the given vectors F1 and F2, the magnitude of the resultant vector is approximately 123 N (Option 3).

To find the direction of the resultant vector, you can use trigonometry. The angle can be found using the tangent of the angle, which is equal to the vertical displacement divided by the horizontal displacement:

Angle = atan(vertical displacement / horizontal displacement)

By substituting the values, the angle of the resultant vector is approximately 23° (Option 3).

To find the magnitude and direction of the resultant vector, F1 + F2, we first need to find the components of both vectors in the x and y directions.

For F1:
Magnitude = 44 N
Angle = 163°

To find the x-component, we use cosine:
F1x = 44 N * cos(163°)

To find the y-component, we use sine:
F1y = 44 N * sin(163°)

Similarly, for F2:
Magnitude = 72 N
Angle = 236°

To find the x-component:
F2x = 72 N * cos(236°)

To find the y-component:
F2y = 72 N * sin(236°)

Now we add the x-components and the y-components separately to find the components of the resultant vector:

Fresultant_x = F1x + F2x
Fresultant_y = F1y + F2y

Next, we find the magnitude of the resultant vector using the Pythagorean theorem:

Magnitude of the resultant vector = √(Fresultant_x² + Fresultant_y²)

Finally, we find the direction of the resultant vector using the inverse tangent function:

Direction = arctan(Fresultant_y / Fresultant_x)

Now, let's calculate the magnitude and direction.

Calculating the components:
F1x = 44 N * cos(163°) = -27.58 N
F1y = 44 N * sin(163°) = 40.42 N

F2x = 72 N * cos(236°) = 37.85 N
F2y = 72 N * sin(236°) = -50.88 N

Calculating the resultant components:
Fresultant_x = -27.58 N + 37.85 N = 10.27 N
Fresultant_y = 40.42 N + (-50.88 N) = -10.46 N

Calculating the magnitude of the resultant vector:
Magnitude of the resultant vector = √(10.27 N² + (-10.46 N)²) = √(105.45 N²) ≈ 10.27 N

Calculating the direction of the resultant vector:
Direction = arctan(-10.46 N / 10.27 N) ≈ -47.105°

Therefore, the magnitude of the resultant vector F1 + F2 is approximately 10.27 N, and the direction is approximately -47.105°.

R = 44N@163o + 72N@236o.

X =Hor.= 44*cos163 + 72*cos236=-82.3 N.
Y =Ver.= 44*sin163 + 72*sin236=-46.8 N.

1. R^2=X^2 + Y^2=(-82.3)^2 + (-46.8)^2 =8963.53.
R = 95 N. = Resultant force.

tanAr = Y/X = -46.8/-82.3 = 0.56865.
Ar = 29.62o = Reference angle.
A = 29.62 + 180 = 210o,CCW. = -150o,CW.