How to Calculate Displacement Vector: Analytical and Graphical Methods

How can you calculate the displacement vector using analytical and graphical methods?

a. What are the magnitudes and angles of each vector involved?

b. What is the addition equation for the vectors?

c. How can you find the resultant vector graphically?

d. How do you find the resultant vector analytically?

e. How to calculate the percentage difference between graphical and analytical magnitudes?

f. How to compare the measured and calculated angles?

Final answer:

To find the displacement (resultant) vector, assign letters to each vector, write an addition equation, draw the vectors graphically, find the resultant vector analytically, and compare the results.

Explanation:

To find the displacement (resultant) vector, we can follow the steps given in the question:

1. Assign letters to each vector. Let's call the first vector A, the second vector B, and the third vector C.
2. Record the magnitudes and angles of each vector. The magnitudes are 40 km, 30 km, and 50 km, and the angles are 0°, 105°, and 240°, respectively.
3. Write the addition equation: A + B + C = R
4. Draw a Cartesian coordinate system and determine the scale.
5. Add the vectors graphically, drawing them tip-to-tail and using a ruler and protractor to ensure accuracy.
6. Label each vector with the appropriate letter, magnitude, and angle.
7. Draw the resultant vector.
8. Use a ruler to determine the magnitude of the resultant vector and a protractor to determine its angle with respect to the positive x-axis.
9. Find the resultant vector analytically by calculating the x and y-components of each vector, finding the x and y-components of the resultant vector, and using these to find the magnitude and angle.
10. Calculate the percentage difference between the magnitudes of the resultant vectors (graphical vs. analytical).
11. Compare the measured and calculated angles.

Following these steps will help us determine the displacement (resultant) vector.

To calculate the displacement vector using both analytical and graphical methods, we need to identify the magnitudes and angles of each vector involved in the scenario:

• Vector A: 40 km (East)
• Vector B: 30 km (15° West of North)
• Vector C: 50 km (30° South of West)

The addition equation for these vectors is A + B + C = R, where R represents the resultant vector.

By drawing the vectors A, B, and C graphically to scale and following the steps mentioned in the question, we can find the magnitude and angle of the resultant vector. This graphical method involves accurately representing each vector and measuring the resultant vector's properties.

On the other hand, the analytical method requires calculating the x and y-components of each vector, combining them to find the resultant vector's components, and determining its magnitude and angle. This method allows for a numerical and systematic approach to finding the displacement vector.

After obtaining the results from both methods, we can calculate the percentage difference between the magnitudes of the resultant vectors and compare the measured and calculated angles to evaluate the accuracy of our calculations.