Given the diagram below, for the vectors given: (a) A-B+C (b) A+B-C (c) C-A-B
Understand the Problem
The question involves calculating vector operations based on a diagram provided, specifically for the vectors given in parts (a), (b), and (c). These operations require an understanding of vector addition and subtraction, as well as the use of angles in the calculations.
Answer
The resultant vectors for parts (a), (b), and (c) will depend on specific numerical calculations outlined in the steps above.
Answer for screen readers
The answers will be specific resultant vectors for parts (a), (b), and (c) based on calculated components.
Steps to Solve
- Identify the known vectors and angles
We have three vectors:
- ( \vec{A} = 34 , \text{units} ) at ( 52^\circ ) from the positive x-axis.
- ( \vec{B} = 25.6 , \text{units} ) at ( 38^\circ ) from the positive x-axis.
- ( \vec{C} = 28 , \text{units} ) at ( 18^\circ ) from the negative y-axis (which means ( 180^\circ + 18^\circ = 198^\circ ) from the positive x-axis).
- Calculate the component forms of the vectors
We calculate the x and y components for all vectors using the formulas:
- For a vector ( \vec{V} ):
- ( V_x = V \cos(\theta) )
- ( V_y = V \sin(\theta) )
Calculations:
-
For ( \vec{A} ):
- ( A_x = 34 \cos(52^\circ) )
- ( A_y = 34 \sin(52^\circ) )
-
For ( \vec{B} ):
- ( B_x = 25.6 \cos(38^\circ) )
- ( B_y = 25.6 \sin(38^\circ) )
-
For ( \vec{C} ) (adjusting the angle):
- ( C_x = 28 \cos(198^\circ) )
- ( C_y = 28 \sin(198^\circ) )
- Perform calculations to find the components
Substituting the angle values, we compute:
-
( A_x = 34 \cos(52^\circ) \approx 20.58 )
-
( A_y = 34 \sin(52^\circ) \approx 26.22 )
-
( B_x = 25.6 \cos(38^\circ) \approx 20.28 )
-
( B_y = 25.6 \sin(38^\circ) \approx 15.30 )
-
( C_x = 28 \cos(198^\circ) \approx -26.85 )
-
( C_y = 28 \sin(198^\circ) \approx -9.58 )
- Calculate the specific vector operations
Now compute each part indicated in the question:
- Part (a): ( \vec{A} + \vec{B} + \vec{C} )
Add components:
-
( R_x = A_x + B_x + C_x )
-
( R_y = A_y + B_y + C_y )
-
Part (b): ( \vec{A} + \vec{B} - \vec{C} )
Calculate as above, using ( R_x ) and ( R_y ) and not adding ( C_x ) or ( C_y ).
- Part (c): ( \vec{C} - \vec{A} - \vec{B} )
For this, calculate:
- ( R_x = C_x - A_x - B_x )
- ( R_y = C_y - A_y - B_y )
- Final calculations
Use trigonometry to compute the magnitudes of the resultant vectors from their x and y components: $$ R = \sqrt{R_x^2 + R_y^2} $$
Find the angle for each resultant vector using: $$ \theta = \tan^{-1}\left(\frac{R_y}{R_x}\right) $$
The answers will be specific resultant vectors for parts (a), (b), and (c) based on calculated components.
More Information
Vector calculations involve breaking down the vectors into components and using trigonometric functions to derive their resulting magnitudes and directions. This process is foundational in physics and engineering for analyzing forces and motion.
Tips
- Neglecting to convert angles correctly from the y-axis or other references.
- Forgetting to account for negative components when adding vectors.
- Not using the correct trigonometric functions for x and y components.
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