https://file.aiquickdraw.com/mathbot/file/1724905338903hdq1q59a.jpg

the image is figure 1: Compute the *x*- and *y*-components of the vectors A⃗ 𝐴→, B⃗ 𝐵→, C⃗ 𝐶→, and D⃗ 𝐷→ in (Figure 1).

**Express your answer in meters.**

Ax,Ay =

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.

Learn how to compute the x- and y-components of vectors A, B, C, and D from Figure 1 using trigonometric principles. This problem involves finding vector components using magnitudes and angles. Suitable for high school students studying trigonometry and vector analysis.

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