- Calculate the components of \(\mathbf{u}\): \(u_x = 5 \cos 75^{\circ} \approx 1.294\), \(u_y = 5 \sin 75^{\circ} \approx 4.830\).
- Calculate the components of \(\mathbf{v}\): \(v_x = 6 \cos 210^{\circ} \approx -5.196\), \(v_y = 6 \sin 210^{\circ} = -3\).
- Add the corresponding components to find the sum: \(x \approx 1.294 - 5.196 = -3.902\), \(y \approx 4.830 - 3 = 1.830\).
- Match the resulting vector \(\langle -3.90, 1.83 \rangle\) to the correct option: \(\boxed{\langle -3.90, 1.83 \rangle}\).
Explanation
- Component Analysis of Vectors u and v
To find the component form of the vector sum \(\mathbf{u} + \mathbf{v}\), we first represent each vector in its component form \(\langle x, y \rangle\), where \(x = r \cos \theta\) and \(y = r \sin \theta\). Here, \(r\) is the magnitude and \(\theta\) is the direction angle.
For vector \(\mathbf{u}\) with magnitude \(|\mathbf{u}| = 5\) and angle \(\theta_u = 75^{\circ}\):
\[u_x = 5 \cos(75^{\circ}) \approx 5 \times 0.2588 = 1.294\]
\[u_y = 5 \sin(75^{\circ}) \approx 5 \times 0.9659 = 4.8296\]
For vector \(\mathbf{v}\) with magnitude \(|\mathbf{v}| = 6\) and angle \(\theta_v = 210^{\circ}\):
\[v_x = 6 \cos(210^{\circ}) = 6 \times (-\frac{\sqrt{3}}{2}) = -3\sqrt{3} \approx -5.196\]
\[v_y = 6 \sin(210^{\circ}) = 6 \times (-\frac{1}{2}) = -3\]
- Summation and Final Identification
Now, we calculate the sum \(\mathbf{u} + \mathbf{v} = \langle u_x + v_x, u_y + v_y \rangle\):
\[x_{sum} = 1.294 + (-5.196) = -3.902\]
\[y_{sum} = 4.8296 + (-3) = 1.8296\]
Comparing these results with the given options \(\langle -3.95, 1.83 \rangle\), \(\langle -3.94, 1.89 \rangle\), \(\langle -3.90, 1.89 \rangle\), and \(\langle -3.90, 1.83 \rangle\), we identify the closest match. Our calculated sum is approximately \(\langle -3.90, 1.83 \rangle\).
Examples
This technique of breaking vectors into components is essential in navigation and physics. For example, if a ship is traveling in one direction and encounters a current pushing it in another direction, pilots add the velocity vectors of the ship and the current to determine the true heading and speed, ensuring the vessel stays on its intended course despite external forces.