1. Vectors (a) A - B + C (b) A + B - C (c) C - A - B
Understand the Problem
The question involves calculating components of vectors using trigonometric functions based on angles given. It focuses on resolving vectors into their horizontal (x) and vertical (y) components and performing vector addition and subtraction.
Answer
The components of the resultant vector \( R \) are approximately \( (38.142, 10.037) \).
Answer for screen readers
The components of the resultant vector ( R ) can be found by plugging in the calculated values.
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Resultant x-component: $$ R_x = (27.122 - 15.183 + 26.203) \approx 38.142 $$
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Resultant y-component: $$ R_y = (20.378 - 19.799 + 9.458) \approx 10.037 $$
Thus, the final answer is: $$ R \approx (38.142, 10.037) $$
Steps to Solve
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Identify Components of Vector A
For vector ( A ):
- The x-component is calculated as: $$ A_x = 34 \cos(38^\circ) $$
- The y-component is calculated as: $$ A_y = 34 \sin(38^\circ) $$
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Identify Components of Vector B
For vector ( B ):
- The x-component is calculated as: $$ B_x = 25.6 \cos(52^\circ) $$
- The y-component is calculated as: $$ B_y = 25.6 \sin(52^\circ) $$
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Identify Components of Vector C
For vector ( C ):
- The x-component is calculated as: $$ C_x = 28 \cos(18^\circ) $$
- The y-component is calculated as: $$ C_y = 28 \sin(18^\circ) $$
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Calculate the Components
Using a calculator:
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For vector ( A ): $$ A_x = 34 \cos(38^\circ) \approx 27.122 $$ $$ A_y = 34 \sin(38^\circ) \approx 20.378 $$
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For vector ( B ): $$ B_x = 25.6 \cos(52^\circ) \approx 15.183 $$ $$ B_y = 25.6 \sin(52^\circ) \approx 19.799 $$
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For vector ( C ): $$ C_x = 28 \cos(18^\circ) \approx 26.203 $$ $$ C_y = 28 \sin(18^\circ) \approx 9.458 $$
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Perform Vector Operations
For the operations given in the problem:
- For ( A - B + C ): $$ (A_x - B_x + C_x, A_y - B_y + C_y) $$
- Substitute the calculated values to find the resultant components.
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Final Resultant Vector Calculation
Calculate the final vector ( R ): $$ R_x = (A_x - B_x + C_x) $$ $$ R_y = (A_y - B_y + C_y) $$
The components of the resultant vector ( R ) can be found by plugging in the calculated values.
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Resultant x-component: $$ R_x = (27.122 - 15.183 + 26.203) \approx 38.142 $$
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Resultant y-component: $$ R_y = (20.378 - 19.799 + 9.458) \approx 10.037 $$
Thus, the final answer is: $$ R \approx (38.142, 10.037) $$
More Information
This calculation illustrates how to resolve vectors into their components and perform vector operations such as addition and subtraction. Understanding these components is crucial in physics and engineering applications where direction and magnitude are important.
Tips
- Using Wrong Angles: Always ensure you are using the correct angle for the sine and cosine functions.
- Confusing Component Signs: Pay attention to the sign of the components based on the quadrant; ensure that the correct positives and negatives are applied.
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