Consider three force vectors ~ F1, ~ F2, and ~ F3.
The vector ~ F1 has magnitude F1 = 86 N and
direction θ1 = 169◦; the vector ~ F2 has magni-
tude F2 = 46 N and direction θ2 = 240◦; and
the vector ~ F3 has magnitude F3 = 45 N and
direction θ3 = 132◦.
All the direction angles θ aremeasured from
the positive x axis: counter-clockwise for θ >
0 and clockwise for θ < 0.
Draw the vectors to scale on a graph to
determine the answer.
What is the magnitude of the resultant vec-
tor k~F k, where ~F = ~ F1 + ~ F2 + ~ F3 ?
^^^ i got 137.89m
now how do i Give the angle in degrees, use coun-
terclockwise as the positive angular direction,
between the limits of −180◦ and +180◦ from
the positive x axis.
What is the direction of this resultant vec-
tor ~F ?
To find the direction of the resultant vector ~F, we can use the following steps:
1. Find the x-components and y-components of each force vector ~F1, ~F2, and ~F3 using the magnitude and direction provided.
For ~F1:
x-component: F1x = F1 * cos(θ1)
y-component: F1y = F1 * sin(θ1)
For ~F2:
x-component: F2x = F2 * cos(θ2)
y-component: F2y = F2 * sin(θ2)
For ~F3:
x-component: F3x = F3 * cos(θ3)
y-component: F3y = F3 * sin(θ3)
2. Calculate the total x-component and y-component of the resultant vector ~F by adding the corresponding components of ~F1, ~F2, and ~F3:
F_total_x = F1x + F2x + F3x
F_total_y = F1y + F2y + F3y
3. Find the magnitude of the total resultant vector ~F:
|~F| = sqrt(F_total_x^2 + F_total_y^2)
4. Find the angle (direction) of the resultant vector ~F:
θ_total = arctan(F_total_y / F_total_x)
Note that the obtained angle will be within the range of -180° to +180°, where positive values indicate a counterclockwise direction from the positive x-axis.
You can now use these steps to calculate the direction of the resultant vector ~F using the values provided for ~F1, ~F2, and ~F3.
To find the direction of the resultant vector ~F, you can use trigonometry and the vector components. Here's how you can determine the angle:
1. Calculate the x-component and y-component of each force vector ~Fi using the magnitudes and angles given. The x-component of a vector ~Fi is given by Fi * cos(theta_i), and the y-component is given by Fi * sin(theta_i).
For ~F1:
x-component of ~F1 = F1 * cos(theta1)
y-component of ~F1 = F1 * sin(theta1)
For ~F2:
x-component of ~F2 = F2 * cos(theta2)
y-component of ~F2 = F2 * sin(theta2)
For ~F3:
x-component of ~F3 = F3 * cos(theta3)
y-component of ~F3 = F3 * sin(theta3)
2. Sum up the x-components and y-components of all three force vectors to find the x-component and y-component of the resultant vector ~F.
x-component of ~F = (x-component of ~F1) + (x-component of ~F2) + (x-component of ~F3)
y-component of ~F = (y-component of ~F1) + (y-component of ~F2) + (y-component of ~F3)
3. Use the calculated x-component and y-component to find the magnitude and direction of the resultant vector ~F using trigonometry. The magnitude of ~F is given by the square root of (x-component^2 + y-component^2).
magnitude of ~F = sqrt((x-component of ~F)^2 + (y-component of ~F)^2)
4. To find the direction of the resultant vector ~F, you can use the arctan function. The direction theta is given by arctan(y-component of ~F / x-component of ~F). However, note that the arctan function will only give you the angle between -90 degrees and +90 degrees. To get the angle between -180 and +180 degrees, you can use the atan2 function, which takes into account the signs of the x-component and y-component.
direction of ~F = atan2(y-component of ~F, x-component of ~F)
Plug in the values for the x-component and y-component from step 2 into the formulas above, and you will find the magnitude and direction of the resultant vector ~F.