To calculate the x- and y-components of each vector, we will use the following formulas:

$A_x = A \cdot \cos(\alpha), \quad A_y = A \cdot \sin(\alpha)$ $B_x = B \cdot \cos(\beta), \quad B_y = B \cdot \sin(\beta)$ $C_x = C \cdot \cos(\gamma), \quad C_y = C \cdot \sin(\gamma)$ $D_x = D \cdot \cos(\delta), \quad D_y = D \cdot \sin(\delta)$

Given values:

• $A = 25.0 \, \text{m}, \, \alpha = 56^\circ$
• $B = 33.4 \, \text{m}, \, \beta = 23^\circ$
• $C = 23.6 \, \text{m}, \, \gamma = 27^\circ$
• $D = 25.4 \, \text{m}, \, \delta = 19^\circ$

1. Vector $\mathbf{A}$:

$A_x = 25.0 \cdot \cos(56^\circ) = 25.0 \cdot 0.559 = 13.975 \, \text{m}$ $A_y = 25.0 \cdot \sin(56^\circ) = 25.0 \cdot 0.829 = 20.725 \, \text{m}$

2. Vector $\mathbf{B}$:

$B_x = 33.4 \cdot \cos(23^\circ) = 33.4 \cdot 0.923 = 30.8 \, \text{m}$ $B_y = 33.4 \cdot \sin(23^\circ) = 33.4 \cdot 0.391 = 13.1 \, \text{m}$

3. Vector $\mathbf{C}$:

$C_x = 23.6 \cdot \cos(27^\circ) = 23.6 \cdot 0.891 = 21.0 \, \text{m}$ $C_y = 23.6 \cdot \sin(27^\circ) = 23.6 \cdot 0.454 = 10.7 \, \text{m}$

4. Vector $\mathbf{D}$:

$D_x = 25.4 \cdot \cos(19^\circ) = 25.4 \cdot 0.945 = 24.0 \, \text{m}$ $D_y = 25.4 \cdot \sin(19^\circ) = 25.4 \cdot 0.325 = 8.3 \, \text{m}$

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Final Results:

1. Vector A:

   • $A_x = 13.98 \, \text{m}$
   • $A_y = 20.73 \, \text{m}$

2. Vector B:

   • $B_x = 30.80 \, \text{m}$
   • $B_y = 13.10 \, \text{m}$

3. Vector C:

   • $C_x = 21.00 \, \text{m}$
   • $C_y = 10.70 \, \text{m}$

4. Vector D:

   • $D_x = 24.00 \, \text{m}$
   • $D_y = 8.30 \, \text{m}$

Let me know if you'd like further clarifications or detailed calculations for any step!

Related Questions:

1. How do these component calculations relate to vector addition?
2. What would happen if the angles were adjusted by 90°?
3. How can you find the resultant vector from the components?
4. What is the significance of the signs in the x- and y-components?
5. How does the angle affect the magnitude of each vector component?

Tip: Always make sure to check the angle's reference direction when calculating vector components.