The three displacement vectors in the drawing have magnitudes of A = 6m B = 6 , and C = 4 . Find the resultant (magnitude and angle) of the three vectors by means of the component method. Express the angle as an angle above the positive or negative x-axis.
The Correct Answer and Explanation is :
To determine the resultant vector from the given displacement vectors A, B, and C with magnitudes 6 m, 6 m, and 4 m respectively, we’ll employ the component method. This approach involves breaking each vector into its horizontal (x) and vertical (y) components, summing these components, and then calculating the magnitude and direction of the resultant vector.
Step 1: Resolve Each Vector into Components
Assuming the angles provided in the diagram are as follows:
- Vector A: 30° above the positive x-axis
- Vector B: 45° below the positive x-axis
- Vector C: 60° above the negative x-axis
We can calculate the components using trigonometric functions:
- Vector A
- ( A_x = A \cos(30^\circ) = 6 \times \frac{\sqrt{3}}{2} = 3\sqrt{3} )
- ( A_y = A \sin(30^\circ) = 6 \times \frac{1}{2} = 3 )
- Vector B
- ( B_x = B \cos(45^\circ) = 6 \times \frac{\sqrt{2}}{2} = 3\sqrt{2} )
- ( B_y = B \sin(45^\circ) = 6 \times \frac{\sqrt{2}}{2} = 3\sqrt{2} )
- Since B is below the x-axis, ( B_y ) is negative: ( B_y = -3\sqrt{2} )
- Vector C
- ( C_x = C \cos(60^\circ) = 4 \times \frac{1}{2} = 2 )
- ( C_y = C \sin(60^\circ) = 4 \times \frac{\sqrt{3}}{2} = 2\sqrt{3} )
- Since C is above the negative x-axis, ( C_x ) is negative: ( C_x = -2 )
Step 2: Sum the Components
Add the x and y components separately to find the components of the resultant vector R:
- ( R_x = A_x + B_x + C_x = 3\sqrt{3} + 3\sqrt{2} – 2 )
- ( R_y = A_y + B_y + C_y = 3 – 3\sqrt{2} + 2\sqrt{3} )
Step 3: Calculate the Magnitude of the Resultant Vector
Use the Pythagorean theorem:
[ R = \sqrt{R_x^2 + R_y^2} ]
Step 4: Determine the Angle of the Resultant Vector
Calculate the angle θ relative to the positive x-axis:
[ \theta = \tan^{-1}\left(\frac{R_y}{R_x}\right) ]
Conclusion
By resolving each vector into its components, summing these components, and applying the Pythagorean theorem and inverse tangent function, we can determine both the magnitude and direction of the resultant vector. This method provides a systematic approach to vector addition, ensuring accurate results when dealing with multiple vectors in different directions.