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Question

Three vectors are shown in the figure. Their magnitudes are given in arbitrary units. Be sure to complete all parts (a) through (g). 

  

a)  Determine the x-component of each of the three vectors using angles in standard position.

b)  Determine the Resultant in the x direction of the sum of three vectors (sum the x-components).

c)  Determine the y-components of each of the three vectors using angles in standard position.

d)  Determine the Resultant in the y direction of the sum of three vectors (sum the y components).

e)  Using the Resultant in the x direction and the Resultant in the y direction, determine magnitude of the Resultant of the sum of the three vectors. (Use the Pythagorean Theorem)

f) Determine the angle (direction) of the Resultant of the sum of the three vectors. The angle should be in standard position, measured counterclockwise from the +x-axis.

g) Write the Resultant Vector in magnitude angle format. (R = R at θ)  

Answer 1

Main thing here is to establish axes. Once this is done the x components will be given by each vector magnitude times the cosine of the angle it takes with the x axis. Similarly the y components are the magnitudes times the sine of the same angle.

Adding components is algebraic with positive being to the right for the x components and up for the y components.

Once the total of x and y components are found, then use the pythagorean theorem to get the magnitude of the resultant. The angle is given by tangent of sum of y components divided by sum of x components

Answer 2

Not having the figure, I am going to assume that the angle for each vector is measured from either the positive x-axis, or the negative x-axis. If any of the angles are measured from the y-axis, the first step would be to determine the angles from either the positive or negative x-axis.

a) Use the cosine function to find the x-component of each vector. For example, vector A will have an x-component of A_x = A cos(θ_A), with the angle measured from the x-axis. If the vector points to the right of the y-axis, then the x-component has a positive value. If the vector points to the left of the y-axis, then the x-component has a negative value.

b) Determine R_x by adding the three x-components together. Remember to include the positive and negative signs that you obtained in part a.

c) Use the sine function to find the y-component of each vector. For example, vector A will have a y-component of A_y = A sin(θ_A), with the angle measured from the x-axis. If the vector points up from the x-axis, then the y-component has a positive value. If the vector points down from the x-axis, then the y-component has a negative value.

d) Determine R_y by adding the three y-components together. Remember to include the positive and negative signs that you obtained in part c.

e) Use the Pythagorean Theorem to find the magnitude of the resultant vector. R = √(R_x² + R_y²).

f) Use the tangent function to find the angle the resultant vector makes with the x-axis. θ_R = tan^-1(R_y/R_x).

If R_x was positive, the angle is measured from the positive x-axis, and if R_x was negative, the angle is measured from the negative x-axis.

If θ_R is positive, the angle is measured counter-clockwise from the x-axis, and if θ_R is negative, the angle is measured clockwise from the x-axis.

If the angle you obtained is not already counter-clockwise from the positive x-axis, then you will need to determine the angle that is counter-clockwise from the positive x-axis (it will be helpful to sketch the x and y axes with the resultant vector shown at the angle you obtained).

g) Your final answers from part e and part f.