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.