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Compute the *x*- and *y*-components of the vectors A⃗ A→, B⃗ B→, C⃗ C→, and D⃗ D→ in the figure

To compute the *x*- and *y*-components of the vectors \(\vec{A}\), \(\vec{B}\), \(\vec{C}\), and \(\vec{D}\), we can use trigonometric functions based on the magnitudes of the vectors and the angles they make with the coordinate axes.

### Components of Vector \(\vec{A}\):
- Magnitude of \(\vec{A} = 8.0 \, \text{m}\)
- Angle with negative \(y\)-axis = \(0^\circ\) (since it's directly downward)

\[
A_x = 8.0 \, \text{m} \times \cos(270^\circ) = 8.0 \, \text{m} \times 0 = 0 \, \text{m}
\]
\[
A_y = 8.0 \, \text{m} \times \sin(270^\circ) = 8.0 \, \text{m} \times (-1) = -8.0 \, \text{m}
\]

### Components of Vector \(\vec{B}\):
- Magnitude of \(\vec{B} = 15.0 \, \text{m}\)
- Angle with positive \(x\)-axis = \(30^\circ\)

\[
B_x = 15.0 \, \text{m} \times \cos(30^\circ) = 15.0 \, \text{m} \times \frac{\sqrt{3}}{2} \approx 12.99 \, \text{m}
\]
\[
B_y = 15.0 \, \text{m} \times \sin(30^\circ) = 15.0 \, \text{m} \times \frac{1}{2} = 7.5 \, \text{m}
\]

### Components of Vector \(\vec{C}\):
- Magnitude of \(\vec{C} = 12.0 \, \text{m}\)
- Angle with negative \(x\)-axis = \(25^\circ\) (which is equivalent to \(180^\circ - 25^\circ = 155^\circ\))

\[
C_x = 12.0 \, \text{m} \times \cos(180^\circ - 25^\circ) = 12.0 \, \text{m} \times (-\cos(25^\circ)) \approx -12.0 \, \text{m} \times 0.9063 \approx -10.88 \, \text{m}
\]
\[
C_y = 12.0 \, \text{m} \times \sin(180^\circ - 25^\circ) = 12.0 \, \text{m} \times \sin(25^\circ) \approx 12.0 \, \text{m} \times 0.4226 \approx 5.07 \, \text{m}
\]

### Components of Vector \(\vec{D}\):
- Magnitude of \(\vec{D} = 10.0 \, \text{m}\)
- Angle with negative \(x\)-axis = \(53^\circ\) (which is equivalent to \(180^\circ - 53^\circ = 127^\circ\))

\[
D_x = 10.0 \, \text{m} \times \cos(127^\circ) = 10.0 \, \text{m} \times (-\cos(53^\circ)) \approx 10.0 \, \text{m} \times (-0.6018) \approx -6.02 \, \text{m}
\]
\[
D_y = 10.0 \, \text{m} \times \sin(127^\circ) = 10.0 \, \text{m} \times \sin(53^\circ) \approx 10.0 \, \text{m} \times 0.7986 \approx 7.99 \, \text{m}
\]

### Summary of Components:
- \(\vec{A}\): \(A_x = 0 \, \text{m}\), \(A_y = -8.0 \, \text{m}\)
- \(\vec{B}\): \(B_x \approx 12.99 \, \text{m}\), \(B_y = 7.5 \, \text{m}\)
- \(\vec{C}\): \(C_x \approx -10.88 \, \text{m}\), \(C_y \approx 5.07 \, \text{m}\)
- \(\vec{D}\): \(D_x \approx -6.02 \, \text{m}\), \(D_y \approx 7.99 \, \text{m}\)

Would you like a more detailed breakdown of any of the components? Here are some related questions that might interest you:

1. How can we determine the resultant vector of all four vectors?
2. What is the magnitude of the resultant vector of \(\vec{A}\) and \(\vec{B}\)?
3. How would the components change if vector \(\vec{B}\) were rotated by an additional \(15^\circ\) counterclockwise?
4. What is the angle between vectors \(\vec{A}\) and \(\vec{D}\)?
5. Can you compute the dot product of vectors \(\vec{C}\) and \(\vec{D}\)?

**Tip:** When dealing with vector components, always ensure you accurately resolve the angles with respect to the coordinate axes to avoid sign errors.

Compute the x- and y-components of the vectors A⃗, B⃗, C⃗, and D⃗ in the figure.

Learn how to calculate the x- and y-components of the vectors A, B, C, and D using trigonometry. This problem demonstrates the decomposition of vectors into their respective components and applies key formulas like Ax = A * cos(θ) and Ay = A * sin(θ). Suitable for high school students in Grades 10-12.

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