A vector `vecA` makes an angle of `20^@ and vecB` makes an angle of `vec110^@` with the X-axis. The magnitude of these vectors are 3 m and 4 m respectively.Find the resultant

Explanation

To solve the problem of finding the resultant vector of `vecA` and `vecB`, we can follow these steps:

### Step 1: Understand the Problem
We are given two vectors:
- `vecA` with a magnitude of 3 m, making an angle of 20° with the X-axis.
- `vecB` with a magnitude of 4 m, making an angle of 110° with the X-axis.

### Step 2: Break the Vectors into Components
We need to find the components of each vector along the X and Y axes.

For `vecA`:
- \( A_x = A \cos(20°) = 3 \cos(20°) \)
- \( A_y = A \sin(20°) = 3 \sin(20°) \)

For `vecB`:
- \( B_x = B \cos(110°) = 4 \cos(110°) \)
- \( B_y = B \sin(110°) = 4 \sin(110°) \)

### Step 3: Calculate the Components
Using a calculator:
- \( A_x = 3 \cos(20°) \approx 3 \times 0.9397 \approx 2.819 \, \text{m} \)
- \( A_y = 3 \sin(20°) \approx 3 \times 0.3420 \approx 1.026 \, \text{m} \)

- \( B_x = 4 \cos(110°) \approx 4 \times (-0.3420) \approx -1.368 \, \text{m} \)
- \( B_y = 4 \sin(110°) \approx 4 \times 0.9397 \approx 3.7588 \, \text{m} \)

### Step 4: Sum the Components
Now, we add the components of the two vectors to find the resultant vector components.

- \( R_x = A_x + B_x = 2.819 + (-1.368) \approx 1.451 \, \text{m} \)
- \( R_y = A_y + B_y = 1.026 + 3.7588 \approx 4.7848 \, \text{m} \)

### Step 5: Calculate the Magnitude of the Resultant Vector
The magnitude of the resultant vector \( R \) can be calculated using the Pythagorean theorem:

\[ R = \sqrt{R_x^2 + R_y^2} = \sqrt{(1.451)^2 + (4.7848)^2} \]

Calculating this gives:
\[ R = \sqrt{2.107 + 22.911} = \sqrt{25.018} \approx 5.0 \, \text{m} \]

### Step 6: Calculate the Direction of the Resultant Vector
To find the angle \( \theta \) that the resultant vector makes with the X-axis, we use the tangent function:

\[ \tan(\theta) = \frac{R_y}{R_x} = \frac{4.7848}{1.451} \]

Calculating this gives:
\[ \theta = \tan^{-1}(3.295) \approx 73.7° \]

### Final Result
The magnitude of the resultant vector is approximately 5.0 m, and it makes an angle of approximately 73.7° with the X-axis.