The problem asks to find the resultant vector $\mathbf{C} = \mathbf{A} + \mathbf{B}$, where:

• Vector $\mathbf{A} = 293$ units at $61.8^\circ$
• Vector $\mathbf{B} = 189$ units at $260.4^\circ$

Steps:

1. Convert the vectors to rectangular form using the equations: $\mathbf{A_x} = A \cos(\theta_A), \quad \mathbf{A_y} = A \sin(\theta_A)$ $\mathbf{B_x} = B \cos(\theta_B), \quad \mathbf{B_y} = B \sin(\theta_B)$

2. Sum the components to find the components of $\mathbf{C}$: $\mathbf{C_x} = \mathbf{A_x} + \mathbf{B_x}, \quad \mathbf{C_y} = \mathbf{A_y} + \mathbf{B_y}$

3. Find the magnitude of $\mathbf{C}$ using the Pythagorean theorem: $|\mathbf{C}| = \sqrt{C_x^2 + C_y^2}$

4. Find the angle $\theta_C$ using: $\theta_C = \tan^{-1}\left(\frac{C_y}{C_x}\right)$

Let's proceed to compute the result.The resultant vector $\mathbf{C}$ has:

• Magnitude: $128.8443$ units
• Angle: $33.9034^\circ$

This matches the first option in the provided multiple-choice answers.

Do you want more details on any step or have questions? Here are some related questions you might find helpful:

1. How do we convert vector components back into polar form?
2. Why do we use trigonometric functions like sine and cosine for vector decomposition?
3. What is the significance of the angle when adding vectors?
4. How do vector magnitudes affect the resultant vector?
5. What would happen if the vectors were perpendicular?

Tip: Always ensure your calculator is set to the correct mode (degrees or radians) when dealing with angles in trigonometric functions.