Learning Goal: To use geometric and component addition of vectors. Four vectors

Question

Learning Goal: To use geometric and component addition of vectors. Four vectors A, B, C, and D are shown (not to scale). Vector A has magnitude 24.5 and acts at an angle of 12.5 degrees with respect to the positive x axis. Vector B has magnitude 12.5 and acts at an angle of 72.3 degrees with respect to the positive x axis. Vector C has magnitude 33.7 and acts at an angle of 141.7 degrees with respect to the positive x axis. Vector D has magnitude 11.0 and acts at an angle of 282.5 degrees with respect to the positive x axis. What are the magnitude and direction of the resultant vector, R, when the parallelogram law is applied to A and B? Express the magnitude to three significant figures. Express the angle to one decimal place, measured counterclockwise from the positive x axis. Separate your answers by a comma. Component addition of vectors What is the resultant vector, R, obtained by adding vectors C and D? Express the magnitude to three significant figures. Express the angle to one decimal place, measured counterclockwise from the positive x axis. Separate your answers by a comma. Addition of more than two vectors Now we wish to add all four vectors. Attempting to do this using the parallelogram method is difficult because that method is optimized for two vectors. It is much easier to use vector components when there are more than two vectors being added. For the vector sum R = A + B + C + D , what are the magnitude and direction of the resultant R? Express the magnitude to three significant figures. Express the angle to one decimal place, measured counterclockwise from the positive x axis. Separate your answers by a comma.

Explanation / Answer

PART A =
A^ = 22.5 * cos(12.5) i^ + 22.5 * sin(12.5) j^

B^ = 13.7 * cos(71.5) i^ + 13.7 * sin(71.5) j^

R^ = A^ + B^

R^ = (22.5 * cos(12.5) + 13.7 * cos(71.5) ) I^ + (22.5 * sin(12.5) + 13.7 * sin(71.5)) J^

R^ = 26.31 i^ + 17.86 j^

|R| = sqrt(26.31^2 + 17.86^2)

|R| = 31.8

direction = tan^-1(17.86/26.31)

direction = 34.2o
Direction = 34.2o counter clockwise from +ve x axis.

PART B =

C^ = -46.9 * sin(52.5) i^ + 46.9 * cos(52.5) j^

D^ = 12 * sin(15.7) i^ - 12 * cos(15.7) j^

R^ = A^ + B^

R^ = (-46.9 * sin(52.5) + 12 * sin(15.7) ) I^ + (46.9 * cos(52.5) - 12 * cos(15.7)) J^

R^ = -33.96 i^ - 16.9 j^

|R| = sqrt(33.96^2 + 16.9^2)

|R| = 37.93

Magnitude , R = 37.93

direction = tan^-1(16.9/33.96)
direction = 180+26.46
Direction = 206.5o counter clockwise from +ve x axis.

PART C =

R^ = R1^ + R2^

R^ = (31.8 * cos(34.2) - 37.93* cos(26.46) ) I^ + (31.8 * sin(34.2) - 37.93* sin(26.46)) J^

R^ = -7.65 i^ - 0.97 j^

|R| = sqrt(7.65^2 + 0.97^2)

|R| = 7.71

Magnitude, R = 7.71

direction = tan^-1(0.97/7.65)

direction = 180 + 7.26

direction = 187.26

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