Math Problem Statement

Resolve the vector given in the indicated figure into its x component and y component.

Solution

To solve the given problem:

We need to resolve the vector A=57A = 57 into its xx-component and yy-component. The angle given is θ=105\theta = 105^\circ, measured counterclockwise from the positive x-axis.

Formula:

  1. The xx-component of the vector is: Ax=Acos(θ)A_x = A \cdot \cos(\theta)
  2. The yy-component of the vector is: Ay=Asin(θ)A_y = A \cdot \sin(\theta)

Steps:

  1. Substitute A=57A = 57 and θ=105\theta = 105^\circ into the formulas.
  2. Compute the values for cos(105)\cos(105^\circ) and sin(105)\sin(105^\circ) using a calculator or trigonometric table.
    • cos(105)=0.2588\cos(105^\circ) = -0.2588
    • sin(105)=0.9659\sin(105^\circ) = 0.9659
  3. Perform the calculations:
    • Ax=57(0.2588)=14.75A_x = 57 \cdot (-0.2588) = -14.75
    • Ay=570.9659=55.06A_y = 57 \cdot 0.9659 = 55.06

Final Answer:

  • Ax=14.8A_x = -14.8 (rounded to the nearest tenth)
  • Ay=55.1A_y = 55.1 (rounded to the nearest tenth)

Let me know if you would like further clarification or a more detailed explanation.


Here are five related questions:

  1. What happens to the components if the angle is in a different quadrant?
  2. How can we verify the results using a vector diagram?
  3. What are the differences between polar and Cartesian coordinate representations?
  4. Can we calculate the magnitude of a vector from its components?
  5. How does the sine and cosine of an angle determine the direction of the components?

Tip: Always visualize the angle relative to the coordinate axes to confirm the signs of the components!

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Math Problem Analysis

Mathematical Concepts

Vector resolution
Trigonometry
Components of a vector

Formulas

Ax = A * cos(θ)
Ay = A * sin(θ)

Theorems

Trigonometric functions of a vector angle

Suitable Grade Level

Grades 10-12