To find the $x$- and $y$-components of the vectors $\vec{A}$, $\vec{B}$, $\vec{C}$, and $\vec{D}$ from the given diagram, we will use trigonometry. For a vector $\vec{V}$ with magnitude $V$ and angle $\theta$ relative to the positive $x$-axis, the components are:

$V_x = V \cos(\theta)$ $V_y = V \sin(\theta)$

Let's calculate the components for each vector:

1. Vector $\vec{A}$:

• Magnitude: $|\vec{A}| = 8.0 \, \text{m}$
• Angle: $\theta_A = 270^\circ$ (or $-90^\circ$) with respect to the positive $x$-axis.

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

2. Vector $\vec{B}$:

• Magnitude: $|\vec{B}| = 15.0 \, \text{m}$
• Angle: $\theta_B = 30^\circ$ with respect to the positive $x$-axis.

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

3. Vector $\vec{C}$:

• Magnitude: $|\vec{C}| = 12.0 \, \text{m}$
• Angle: $\theta_C = 180^\circ - 25^\circ = 155^\circ$ with respect to the positive $x$-axis.

$C_x = 12.0 \times \cos(155^\circ) = 12.0 \times (-0.9063) = -10.88 \, \text{m}$ $C_y = 12.0 \times \sin(155^\circ) = 12.0 \times 0.4226 = 5.07 \, \text{m}$

4. Vector $\vec{D}$:

• Magnitude: $|\vec{D}| = 10.0 \, \text{m}$
• Angle: $\theta_D = 53^\circ$ with respect to the positive $x$-axis.

$D_x = 10.0 \times \cos(53^\circ) = 10.0 \times 0.6018 = 6.02 \, \text{m}$ $D_y = 10.0 \times \sin(53^\circ) = 10.0 \times 0.7986 = 7.99 \, \text{m}$

Summary:

• $A_x = 0 \, \text{m}, \, A_y = -8.0 \, \text{m}$
• $B_x = 12.99 \, \text{m}, \, B_y = 7.5 \, \text{m}$
• $C_x = -10.88 \, \text{m}, \, C_y = 5.07 \, \text{m}$
• $D_x = 6.02 \, \text{m}, \, D_y = 7.99 \, \text{m}$

Would you like more details or have any other questions?

Here are some follow-up questions:

1. How can you find the resultant vector from these components?
2. What is the significance of the angles used in these calculations?
3. How would the components change if vector $A$ was rotated by 90 degrees?
4. What is the process to calculate the magnitude of a vector from its components?
5. How do you add two vectors given in component form?

Tip: Always check the quadrant of the angle to determine the correct signs for the components.