The three displacement vectors in the drawing have magnitudes of A = 5.73 m, B = 6.48 m, and C = 3.12 m. Find the resultant
((a) magnitude and (b) directional angle) of the three vectors by means of the component method. Express the directional angle as an angle above the positive or negative x axis.
I need to know the angles.
To find the resultant vector using the component method, follow these steps:
Step 1: Resolve each vector into its x and y components.
- For vector A with magnitude A = 5.73 m, determine its x-component (Ax) and y-component (Ay).
- For vector B with magnitude B = 6.48 m, determine its x-component (Bx) and y-component (By).
- For vector C with magnitude C = 3.12 m, determine its x-component (Cx) and y-component (Cy).
Step 2: Add the x-components and y-components separately.
- Add the x-components: Rx = Ax + Bx + Cx
- Add the y-components: Ry = Ay + By + Cy
Step 3: Calculate the magnitude of the resultant vector.
- Use the Pythagorean theorem: R = sqrt(Rx^2 + Ry^2)
Step 4: Calculate the directional angle.
- Use the arctangent function: θ = atan(Ry / Rx)
- Consider the quadrant in which the angle lies to determine the correct angle above the x-axis or below the x-axis.
Let's calculate the resultant vector:
Step 1: Resolving the vectors:
- Let's assume vector A is at an angle (α) with respect to the positive x-axis.
Therefore, Ax = A * cos(α) and Ay = A * sin(α).
- Similarly, for vector B at an angle (β): Bx = B * cos(β) and By = B * sin(β).
- For vector C at an angle (γ): Cx = C * cos(γ) and Cy = C * sin(γ).
Step 2: Adding the x-components and y-components:
Rx = Ax + Bx + Cx
Ry = Ay + By + Cy
Step 3: Magnitude of the resultant vector:
R = sqrt(Rx^2 + Ry^2)
Step 4: Directional angle:
θ = atan(Ry / Rx)
Simply plug in the values of A, B, C, α, β, and γ into the equations to find the magnitude and directional angle of the resultant vector.