Find the magnitude and direction of the resultant R of the three vectors shown in Figure 1.3.

Question

Find the magnitude and direction of the resultant R of the three vectors shown in Figure 1.3. The vectors have the following magnitudes: A = 5.0, B = 7.9, and C = 8.0. Express the direction of the vector sum by specifying the angle it makes with the positive x-axis, with the counterclockwise angles taken to be positive.

Answer 1

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.

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.

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

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.

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

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

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 measured from the positive x-axis, then you will need to determine the angle that is measured 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).