To find the magnitude of the resultant vector $R$ from the sum of two vectors $A$ and $B$, we first need to break down each vector into its components along the $x$-axis and $y$-axis, and then sum these components.

Step 1: Components of vector $A$

The magnitude of vector $A$ is 80.3 N and the angle is 74.0° above the positive $x$-axis. The components are:

• $A_x = A \cdot \cos(\theta_A) = 80.3 \cdot \cos(74.0^\circ)$
• $A_y = A \cdot \sin(\theta_A) = 80.3 \cdot \sin(74.0^\circ)$

Step 2: Components of vector $B$

The magnitude of vector $B$ is 83.1 N and the angle is 67.5° above the negative $x$-axis. Since it is on the negative $x$-axis, the components will be:

• $B_x = B \cdot \cos(\theta_B) = 83.1 \cdot \cos(180^\circ - 67.5^\circ) = 83.1 \cdot \cos(112.5^\circ)$
• $B_y = B \cdot \sin(\theta_B) = 83.1 \cdot \sin(112.5^\circ)$

Step 3: Sum of the components

To find the resultant vector $R$, sum the components:

• $R_x = A_x + B_x$
• $R_y = A_y + B_y$

Step 4: Magnitude of the resultant vector

The magnitude of the resultant vector is given by:

$|R| = \sqrt{R_x^2 + R_y^2}$

Now, let's calculate these values.The magnitude of the resultant vector is approximately $154.3 \, \text{N}$.

Would you like more details or have any further questions?

Here are some related questions:

1. How do you calculate the direction of the resultant vector?
2. Can you explain the significance of vector components in physics?
3. What happens if both vectors lie along the same axis?
4. How would the problem change if one of the angles was negative?
5. Can vectors have a negative magnitude?

Tip: When working with vectors, always break them into their components for easier calculations.