Add the following vectors using components.
3 N in a direction 15° south of west and 4 N in a direction 12° east of south
<-4cos15°+4sin12°, -4sin15°-3cos12°> = <-3.03,-3.97>
or approximately <-3,-4> = 5 @ S36.8°W
All angles are measured CCW from +X-axis.
Fr = 3N[195o] + 4N[282o] .
X = 3*Cos195 + 4*Cos282 = -2.07 N.
Y = 3*sin195 + 4*sin282 = -4.69 N.
Fr = sqrt(X^2 + Y^2) = 5.13 N. = Resultant force.
TanA = Y/X, A = 66.2o S. of W. = 246.2o CCW.
To add the vectors using components, we have to break each vector into its horizontal and vertical components.
1. Vector 1: 3 N at 15° south of west
- Horizontal component: 3 N * cos(15°) = 2.877 N, directed to the west
- Vertical component: 3 N * sin(15°) = -0.754 N, directed south
2. Vector 2: 4 N at 12° east of south
- Horizontal component: 4 N * cos(-12°) = 3.941 N, directed east
- Vertical component: 4 N * sin(-12°) = -0.835 N, directed south
Now, we add the horizontal and vertical components separately:
Horizontal component: 2.877 N + 3.941 N = 6.818 N, directed east
Vertical component: -0.754 N + (-0.835 N) = -1.589 N, directed south
Finally, we combine the components to find the resultant vector:
Resultant vector: √(horizontal component^2 + vertical component^2)
= √((6.818 N)^2 + (-1.589 N)^2)
≈ √(46.530 N^2 + 2.521 N^2)
≈ √49.051 N^2
≈ 7 N
Therefore, the resultant vector is approximately 7 N, directed to the east of south.
To add the vectors using components, we need to break down each vector into its horizontal and vertical components.
Let's start with the vector 3 N in a direction 15° south of west:
- The horizontal component can be found by multiplying the magnitude (3 N) by the cosine of the angle (15° south of west). The angle is south of west, so it is 180° - 15° = 165°. Using cosine: horizontal component = 3 N * cos(165°).
- The vertical component can be found by multiplying the magnitude (3 N) by the sine of the angle (15° south of west). The angle is south of west, so it is 180° - 15° = 165°. Using sine: vertical component = 3 N * sin(165°).
Now, let's find the components for the vector 4 N in a direction 12° east of south:
- The horizontal component can be found by multiplying the magnitude (4 N) by the cosine of the angle (12° east of south). The angle is south, so the horizontal component is in the negative direction. Using cosine: horizontal component = 4 N * cos(-12°).
- The vertical component can be found by multiplying the magnitude (4 N) by the sine of the angle (12° east of south). The angle is south, so the vertical component is in the negative direction. Using sine: vertical component = 4 N * sin(-12°).
Now that we have the components for both vectors, we can add them together separately in the horizontal and vertical directions. The horizontal component is the sum of the horizontal components, and the vertical component is the sum of the vertical components.
Add the horizontal components: horizontal component = (3 N * cos(165°)) + (4 N * cos(-12°)).
Add the vertical components: vertical component = (3 N * sin(165°)) + (4 N * sin(-12°)).
Finally, we can find the magnitude and direction of the resultant vector using the horizontal and vertical components. The magnitude can be found using the Pythagorean theorem: magnitude = sqrt((horizontal component)^2 + (vertical component)^2).
The direction can be found using the inverse tangent function: direction = atan(vertical component / horizontal component).
Evaluate these formulas to get the final result.