https://file.aiquickdraw.com/mathbot/file/1729902790434n0uguuc6.jpg

To find the direction of the vector sum \(\vec{A} + \vec{B}\), we need to break down each vector into its horizontal (x) and vertical (y) components, add these components, and then determine the angle of the resultant vector.

### Step-by-Step Solution

1. **Resolve Vector A into Components:**
   - Magnitude of \(\vec{A}\) = 17.6 m
   - Direction of \(\vec{A}\) = 32.8°
   \[
   A_x = 17.6 \cos(32.8^\circ)
   \]
   \[
   A_y = 17.6 \sin(32.8^\circ)
   \]

2. **Resolve Vector B into Components:**
   - Magnitude of \(\vec{B}\) = 236 m
   - Direction of \(\vec{B}\) = 63.9°
   \[
   B_x = 236 \cos(63.9^\circ)
   \]
   \[
   B_y = 236 \sin(63.9^\circ)
   \]

3. **Calculate the Resultant Components:**
   - Sum the x-components and y-components:
   \[
   R_x = A_x + B_x
   \]
   \[
   R_y = A_y + B_y
   \]

4. **Determine the Direction of the Resultant Vector:**
   - The direction angle \(\theta\) of \(\vec{R}\) with respect to the positive x-axis can be found using:
   \[
   \theta = \tan^{-1} \left(\frac{R_y}{R_x}\right)
   \]

Let's calculate each of these steps.The direction of the resultant vector \(\vec{A} + \vec{B}\) is approximately \(61.8^\circ\) with respect to the positive x-axis.

### Summary of Calculations:
- Resultant x-component, \( R_x = 118.62 \) m
- Resultant y-component, \( R_y = 221.47 \) m
- Direction of \(\vec{A} + \vec{B}\): \( \theta = 61.8^\circ \)

Would you like further details on any step?

### Related Questions
1. How can you calculate the magnitude of the resultant vector \(\vec{A} + \vec{B}\)?
2. What if \(\vec{A}\) or \(\vec{B}\) had a negative angle? How would that affect the result?
3. How do you find the angle between two vectors given their magnitudes and directions?
4. What are some practical applications of vector addition in physics?
5. How would the result change if \(\vec{A}\) and \(\vec{B}\) were in opposite directions?

### Tip:
Always convert angles to radians when using trigonometric functions in calculations involving angles, unless your calculator or software specifies otherwise.

Find the direction of the vector sum \( \vec{A} + \vec{B} \). Vector A is 17.6 m long in a 32.8° direction, and Vector B is 236 m long in a 63.9° direction.

Learn how to find the direction of the vector sum \( \vec{A} + \vec{B} \), with Vector A at 17.6 m in a 32.8° direction and Vector B at 236 m in a 63.9° direction. This solution covers vector addition using trigonometric functions and angle calculations, suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation