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Bot GPT-4o To find the direction of the vector sum \(\mathbf{a} + \mathbf{b}\), we can follow these steps:
1. **Break down the vectors into their components:**
- Vector \(\mathbf{a}\):
- Magnitude \(a = 3250\) m
- Angle \(\theta_a = 38.4^\circ\)
- Components:
\[
a_x = a \cos \theta_a = 3250 \cos 38.4^\circ
\]
\[
a_y = a \sin \theta_a = 3250 \sin 38.4^\circ
\]
- Vector \(\mathbf{b}\):
- Magnitude \(b = 1822\) m
- Angle \(\theta_b = 58.8^\circ\)
- Components:
\[
b_x = b \cos \theta_b = 1822 \cos 58.8^\circ
\]
\[
b_y = b \sin \theta_b = 1822 \sin 58.8^\circ
\]
2. **Calculate each component:**
- For \(\mathbf{a}\):
\[
a_x = 3250 \cos 38.4^\circ \approx 2550.35 \text{ m}
\]
\[
a_y = 3250 \sin 38.4^\circ \approx 2022.38 \text{ m}
\]
- For \(\mathbf{b}\):
\[
b_x = 1822 \cos 58.8^\circ \approx 944.77 \text{ m}
\]
\[
b_y = 1822 \sin 58.8^\circ \approx 1552.37 \text{ m}
\]
3. **Sum the components:**
- Sum of the \(x\)-components:
\[
R_x = a_x + b_x = 2550.35 + 944.77 \approx 3495.12 \text{ m}
\]
- Sum of the \(y\)-components:
\[
R_y = a_y + b_y = 2022.38 + 1552.37 \approx 3574.75 \text{ m}
\]
4. **Find the direction of the resulting vector \(\mathbf{R}\):**
- The direction \(\theta_R\) of the resultant vector can be found using the arctangent of the ratio of the sum of the \(y\)-components to the sum of the \(x\)-components:
\[
\theta_R = \tan^{-1}\left(\frac{R_y}{R_x}\right)
\]
5. **Calculate the direction:**
\[
\theta_R = \tan^{-1}\left(\frac{3574.75}{3495.12}\right) \approx \tan^{-1}(1.0227) \approx 45.7^\circ
\]
Therefore, the direction of the vector sum \(\mathbf{a} + \mathbf{b}\) is approximately \(45.7^\circ\).