Two vectors are given below. Determine the magnitude in N of the resultant sum of the two vectors.

 

A = 80.3 N, 74.0 degrees above the positive x-axis

B = 83.1 N, 67.5 degrees above the negative x-axis



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.

Two vectors are given below. Determine the magnitude in N of the resultant sum of the two vectors. A = 80.3 N, 74.0 degrees above the positive x-axis B = 83.1 N, 67.5 degrees above the negative x-axis

Solve a vector addition problem where vector A is 80.3 N at 74.0 degrees above the x-axis, and vector B is 83.1 N at 67.5 degrees above the negative x-axis. Learn how to use trigonometry to break down vectors and find the magnitude of the resultant using the Pythagorean Theorem. Suitable for Grades 10-12.

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