Multiple-Concept Example 9 provides background pertinent to this problem. The magnitudes of the four displacement vectors shown in the
drawing are A = 16,0 m, B = 11.0 m, C = 12.0 m, and D = 26.0 m
Determine the magnitude and directional angle for the resultant tha occurs when these vectors are added together
To find the magnitude and directional angle of the resultant vector, we can use the Pythagorean theorem and the tangent function.
First, let's add the vectors together:
Resultant vector = A + B + C + D
Resultant magnitude:
Magnitude = sqrt(A^2 + B^2 + C^2 + D^2)
Magnitude = sqrt((16.0)^2 + (11.0)^2 + (12.0)^2 + (26.0)^2)
Magnitude ≈ sqrt(256.0 + 121.0 + 144.0 + 676.0)
Magnitude ≈ sqrt(1197.0)
Magnitude ≈ 34.6 m
To find the directional angle, we need to find the components of the resultant vector in the x-axis and y-axis.
Component in the x-axis:
Rx = Ax + Bx + Cx + Dx
Rx = 16.0 m + 11.0 m + 12.0 m + 26.0 m
Rx = 65.0 m
Component in the y-axis:
Ry = Ay + By + Cy + Dy
Ry = 0 m + 0 m + 0 m + 0 m
Ry = 0 m
Now, we can use the tangent function to find the directional angle:
Directional angle = tan^(-1)(Ry / Rx)
Directional angle = tan^(-1)(0 / 65.0)
Directional angle ≈ tan^(-1)(0)
Directional angle ≈ 0°
Therefore, the magnitude of the resultant vector is approximately 34.6 m and the directional angle is 0° (or along the positive x-axis).
To determine the magnitude and directional angle of the resultant of these vectors, we need to perform vector addition. We can use the Pythagorean theorem and trigonometric functions to find the magnitude and directional angle:
1. Start by drawing the vectors in a coordinate system, noting their magnitudes:
A = 16.0 m (pointing in the positive x-direction),
B = 11.0 m (pointing in the negative y-direction),
C = 12.0 m (pointing in the positive y-direction), and
D = 26.0 m (pointing in the negative x-direction).
2. Add the vectors by components:
Rx = Ax + Dx = 16.0 m + (-26.0 m) = -10.0 m,
Ry = By + Cy = (-11.0 m) + 12.0 m = 1.0 m,
Note: The x-component is negative because vector D is pointing in the negative x-direction.
3. Use the Pythagorean theorem to find the magnitude of the resultant:
R = √(Rx^2 + Ry^2) = √((-10.0 m)^2 + (1.0 m)^2) = √(100.0 m^2 + 1.0 m^2) ≈ √101.0 m ≈ 10.05 m
4. Use trigonometric functions to find the directional angle (θ) of the resultant:
θ = tan^(-1)(Ry / Rx) = tan^(-1)(1.0 m / -10.0 m) ≈ -5.71°
Note: Since Rx is negative, the angle is measured counterclockwise from the positive x-direction, giving a negative result.
Therefore, the magnitude of the resultant is approximately 10.05 m, and the directional angle is approximately -5.71°.
To find the magnitude and directional angle of the resultant vector, we need to add the given displacement vectors together. Let's go step by step:
Step 1: Start by drawing a rough sketch of the vectors A, B, C, and D. Make sure to show their directions and magnitudes accurately.
Step 2: Define a coordinate system to work with. For simplicity, let's assume the x-axis is horizontal and the y-axis is vertical.
Step 3: Resolve each vector into its x and y components. To do this, use trigonometry.
For vector A, since it has a magnitude of 16.0 m and is purely horizontal, its x-component is 16.0 and y-component is 0.
For vector B, since it has a magnitude of 11.0 m and is purely vertical, its x-component is 0 and y-component is 11.0.
For vector C, let's assume it makes an angle θ with the positive x-axis. To find its x and y components, we can use trigonometry. Given the magnitude C = 12.0 m, we have:
C_x = C * cos(θ)
C_y = C * sin(θ)
For vector D, let's assume it makes an angle φ with the positive x-axis. We can use the same process as vector C to find its x and y components.
D_x = D * cos(φ)
D_y = D * sin(φ)
Step 4: Now, add up the x-components and y-components separately.
Resultant_x = A_x + B_x + C_x + D_x
Resultant_y = A_y + B_y + C_y + D_y
Step 5: Calculate the magnitude of the resultant vector using the Pythagorean theorem:
Resultant_magnitude = sqrt(Resultant_x^2 + Resultant_y^2)
Step 6: Finally, calculate the directional angle θ_r of the resultant vector using the inverse tangent function:
θ_r = arctan(Resultant_y / Resultant_x)
Now that you have the value for Resultant_magnitude and θ_r, you can find the magnitude and directional angle of the resultant vector when the given displacement vectors are added together.