Questions: Physics Question: Sum of 2 vectors A+B=?

Physics Question: Sum of 2 vectors A+B=?
Transcript text: Physics Question: Sum of 2 vectors \[ A+B=\text { ? } \]
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Solution

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Solution Steps

Step 1: Identify the Vectors and Their Components
  • Vector A: 8 m at 25° below the x-axis.
  • Vector B: 10 m at 30° above the x-axis.
Step 2: Calculate the Components of Vector A
  • \( A_x = 8 \cos(25°) \)
  • \( A_y = -8 \sin(25°) \) (negative because it's below the x-axis)
Step 3: Calculate the Components of Vector B
  • \( B_x = 10 \cos(30°) \)
  • \( B_y = 10 \sin(30°) \) (positive because it's above the x-axis)
Step 4: Sum the Components
  • \( R_x = A_x + B_x \)
  • \( R_y = A_y + B_y \)
Step 5: Calculate the Magnitude and Direction of the Resultant Vector
  • Magnitude: \( R = \sqrt{R_x^2 + R_y^2} \)
  • Direction: \( \theta = \tan^{-1}\left(\frac{R_y}{R_x}\right) \)

Final Answer

  1. Calculate \( A_x \) and \( A_y \):

    • \( A_x = 8 \cos(25°) \approx 7.25 \)
    • \( A_y = -8 \sin(25°) \approx -3.38 \)

  2. Calculate \( B_x \) and \( B_y \):

    • \( B_x = 10 \cos(30°) \approx 8.66 \)
    • \( B_y = 10 \sin(30°) \approx 5.00 \)

  3. Sum the components:

    • \( R_x = 7.25 + 8.66 \approx 15.91 \)
    • \( R_y = -3.38 + 5.00 \approx 1.62 \)

  4. Calculate the magnitude and direction:

    • \( R = \sqrt{15.91^2 + 1.62^2} \approx 16.00 \)
    • \( \theta = \tan^{-1}\left(\frac{1.62}{15.91}\right) \approx 5.82° \) above the x-axis
Final Answer

The resultant vector \( \mathbf{R} \) has a magnitude of approximately 16.00 m and is directed at an angle of approximately 5.82° above the x-axis.

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