1. Vectors (a) A - B + C (b) A + B - C (c) C - A - B

Steps to Solve

1. Resolve Vectors into Components

For each vector, we will resolve it into its x and y components using the formulas provided.

• For vector A:

  • ( A_x = 34.0 \cos(38^\circ) )
  • ( A_y = 34.0 \sin(38^\circ) )

• For vector B:

  • ( B_x = 25.6 \cos(52^\circ) )
  • ( B_y = 25.6 \sin(52^\circ) )

• For vector C:

  • ( C_x = 28.0 \cos(180^\circ) )
  • ( C_y = 28.0 \sin(180^\circ) )

2. Calculating Individual Components

Now, we can calculate the actual values for each component.

• For vector A:

  • ( A_x = 34.0 \cos(38^\circ) \approx 27.0 )
  • ( A_y = 34.0 \sin(38^\circ) \approx 20.6 )

• For vector B:

  • ( B_x = 25.6 \cos(52^\circ) \approx 15.4 )
  • ( B_y = 25.6 \sin(52^\circ) \approx 19.8 )

• For vector C:

  • ( C_x = 28.0 \cos(180^\circ) = -28.0 )
  • ( C_y = 28.0 \sin(180^\circ) = 0 )

3. Performing Vector Calculations (Example a: A - B + C)

We now perform the specified operations on the vectors.

For ( A - B + C ):

• Resultant components:
  • ( R_x = A_x - B_x + C_x )
  • ( R_y = A_y - B_y + C_y )

Substituting the values:

• ( R_x = 27.0 - 15.4 - 28.0 )
• ( R_y = 20.6 - 19.8 + 0 )

4. Calculating Resultant Components

Now, calculate ( R_x ) and ( R_y ):

• ( R_x = 27.0 - 15.4 - 28.0 \approx -16.4 )
• ( R_y = 20.6 - 19.8 \approx 0.8 )

5. Finding the Resultant Magnitude and Angle

Finally, we can find the magnitude and angle of the resultant vector:

• Magnitude: $$ R = \sqrt{R_x^2 + R_y^2} $$ Substituting, $$ R = \sqrt{(-16.4)^2 + (0.8)^2} $$

• Angle: $$ \theta = \tan^{-1}\left(\frac{R_y}{R_x}\right) $$

6. Calculate the Final Values for Magnitude and Angle

Calculating these values gives:

• ( R = \sqrt{268.96 + 0.64} \approx \sqrt{269.6} \approx 16.4 )
• ( \theta = \tan^{-1}\left(\frac{0.8}{-16.4}\right) )