https://file.aiquickdraw.com/mathbot/file/1727345617746idybtnki.jpg

The problem in the image asks for **vector addition using the analytical method**, and provides three vectors:

- **Vector A**: 60 N, \( 245^\circ \) from the +X-axis
- **Vector B**: 70 N, \( 75^\circ \) north of east
- **Vector C**: 95 N, \( 87^\circ \) west of north

Let's break this down step-by-step.

### 1. Convert each vector to its components:

#### Vector A
- Magnitude: 60 N
- Angle: \( 245^\circ \) from the +X-axis (standard position; i.e., counterclockwise from the positive X-axis)
  - This angle places the vector in the third quadrant, so both components will be negative.
  
Components:
\[
A_x = 60 \times \cos(245^\circ)
\]
\[
A_y = 60 \times \sin(245^\circ)
\]

#### Vector B
- Magnitude: 70 N
- Angle: \( 75^\circ \) north of east
  - This means \( 75^\circ \) measured counterclockwise from the positive X-axis.
  
Components:
\[
B_x = 70 \times \cos(75^\circ)
\]
\[
B_y = 70 \times \sin(75^\circ)
\]

#### Vector C
- Magnitude: 95 N
- Angle: \( 87^\circ \) west of north
  - This is effectively \( 90^\circ - 87^\circ = 3^\circ \) west of the positive Y-axis, or \( 90^\circ + 3^\circ = 93^\circ \) counterclockwise from the positive X-axis.
  
Components:
\[
C_x = 95 \times \cos(93^\circ)
\]
\[
C_y = 95 \times \sin(93^\circ)
\]

### 2. Add the components
Sum the X and Y components of the three vectors to find the resultant vector components:
\[
R_x = A_x + B_x + C_x
\]
\[
R_y = A_y + B_y + C_y
\]

### 3. Resultant vector magnitude and direction
The magnitude \( R \) of the resultant vector can be found using:
\[
R = \sqrt{R_x^2 + R_y^2}
\]
And the direction \( \theta \) is given by:
\[
\theta = \tan^{-1}\left(\frac{R_y}{R_x}\right)
\]

I will calculate this step-by-step for you.Here are the detailed results for each vector and the resultant:

### Components:
- **Vector A**: 
  - \( A_x = -25.4 \, \text{N} \)
  - \( A_y = -54.4 \, \text{N} \)
  
- **Vector B**:
  - \( B_x = 18.1 \, \text{N} \)
  - \( B_y = 67.6 \, \text{N} \)

- **Vector C**:
  - \( C_x = -5.0 \, \text{N} \)
  - \( C_y = 94.9 \, \text{N} \)

### Resultant Vector:
- **Resultant Components**:
  - \( R_x = -12.2 \, \text{N} \)
  - \( R_y = 108.1 \, \text{N} \)
  
- **Magnitude**: \( 108.8 \, \text{N} \)
- **Direction**: \( 96.4^\circ \) (counterclockwise from the positive X-axis)

Would you like more details on any of these steps, or have any other questions?

Here are 5 questions for further exploration:
1. How do you calculate the horizontal component of a vector from its magnitude and angle?
2. What is the importance of breaking vectors into components in vector addition?
3. How would the result change if one vector's direction were reversed?
4. Can you verify the direction of the resultant vector using a graphical method?
5. How can vector addition be applied in real-world physics problems?

**Tip**: Always ensure angles are correctly converted from directional descriptions (e.g., "north of east") to standard angles relative to the X-axis.

Vector addition using analytical method. Vector A = 60N, 245 degrees from +X-axis. Vector B = 70N, 75 degrees north of east. Vector C = 95N, 87 degrees W of north. What is the horizontal component of vector B?

Learn how to solve vector addition using the analytical method with detailed examples. Vectors A, B, and C are decomposed into components, and the resultant vector is calculated. Suitable for Grades 11-12 and introductory college physics.

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