Question

Add the following vectors using head-to-tail method and verify your results using the component method.

1. Vector A has a magnitude of 4.0 cms at an angle of 30 degrees with the positive X-axis; vector B has a magnitude of 3.0 cms at 90 degrees with + X-axis; vector C has a magnitude of 5.0 cms at 120 degrees with + X-axis.
2. In addition to vectors A, B, and C in problem # 1, there is a vector D with a magnitude of 6.0 cms at an angle of 210 degrees with +X-axis.




Note: Use a graph paper to show your work for problem number 1 and 2 using the head-to-tail method.

Do the component method analysis of both of these problems using the component method of adding vectors. Show your work step by step by taking the x and y components of all the vectors; adding all the x and y components together and then finding the resultant vector magnitude and direction knowing x and y are perpendicular to each other.

Compare the results of the graphical analysis as well as the component method and find the percent error using the component method as standard result
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Answer 1

Answer:

See attached image for the requested graphs.

Part 1 : Vector sum is about 9.4 cm long in the graph, from components: about 9.35 cm long. Percent difference = 0.5%

Part 2: Vector sum is about 7.6 cm long in the graph and from components: 7.57 cm long. Percent difference = 0.4%

Explanation:

Part 1.

The graphical addition of the three vectors A, B and C gives a vector sum of approximately 9.4 cm at an angle of about 84 degrees

The component form addition is

Ax + Bx + Cx  = 3.5 + 0 + (-2.5)  = 1

Ay + By + Cy  = 2.0 + 3 + 4.3  = 9.3

Magnitude: $\sqrt{1^2+9.3^2} \approx 9.35\,\,cm$

The percent difference is: (9.4-9.35) * 100 /9.35 = 0.5%

Part 2.

The graphical addition of the four vectors (A, B, C, and D) measures approximately  7.6 cm at an angle of about 124 degrees.

The component form addition is

Ax + Bx + Cx + Dx = 3.5 + 0 + (-2.5) + (-5.2) = -4.2

Ay + By + Cy + Dy = 2.0 + 3 + 4.3 + (-3) = 6.3

Magnitude = $\sqrt{(-4.2)^2+6.3^2} \approx 7.57\,\,cm$

The percent difference is: (7.6-7.57) * 100 /7.57 = 0.4%