A vector `vec(A)` and `vec(B)` make angles of `20^(@)` and `110^(@)` respectively with the X-axis. The magnitudes of these vectors are `5m` and `12m` respectively. Find their resultant vector.

Explanation

To find the resultant vector of `vec(A)` and `vec(B)`, we will follow these steps:

### Step 1: Identify the given information
- Magnitude of vector `vec(A) = 5 m`
- Angle of vector `vec(A) = 20°` with the X-axis
- Magnitude of vector `vec(B) = 12 m`
- Angle of vector `vec(B) = 110°` with the X-axis

### Step 2: Calculate the components of each vector
We can resolve each vector into its X and Y components using trigonometric functions.

For vector `vec(A)`:
- \( A_x = A \cdot \cos(\theta_A) = 5 \cdot \cos(20°) \)
- \( A_y = A \cdot \sin(\theta_A) = 5 \cdot \sin(20°) \)

For vector `vec(B)`:
- \( B_x = B \cdot \cos(\theta_B) = 12 \cdot \cos(110°) \)
- \( B_y = B \cdot \sin(\theta_B) = 12 \cdot \sin(110°) \)

### Step 3: Calculate the components
Using the values of cosine and sine for the angles:
- \( A_x = 5 \cdot \cos(20°) \approx 5 \cdot 0.9397 \approx 4.6985 \)
- \( A_y = 5 \cdot \sin(20°) \approx 5 \cdot 0.3420 \approx 1.7100 \)

- \( B_x = 12 \cdot \cos(110°) \approx 12 \cdot (-0.3420) \approx -4.1040 \)
- \( B_y = 12 \cdot \sin(110°) \approx 12 \cdot 0.9397 \approx 11.2764 \)

### Step 4: Sum the components to find the resultant vector
Now we can find the resultant vector components:
- \( R_x = A_x + B_x = 4.6985 - 4.1040 \approx 0.5945 \)
- \( R_y = A_y + B_y = 1.7100 + 11.2764 \approx 12.9864 \)

### Step 5: Calculate the magnitude of the resultant vector
Using the Pythagorean theorem:
- \( R = \sqrt{R_x^2 + R_y^2} = \sqrt{(0.5945)^2 + (12.9864)^2} \)
- \( R \approx \sqrt{0.3534 + 168.6774} \)
- \( R \approx \sqrt{169.0308} \approx 13.0 \, m \)

### Step 6: Find the direction of the resultant vector
To find the angle \( \theta_R \) of the resultant vector with respect to the X-axis, we use:
- \( \theta_R = \tan^{-1}\left(\frac{R_y}{R_x}\right) = \tan^{-1}\left(\frac{12.9864}{0.5945}\right) \)

Calculating this gives:
- \( \theta_R \approx \tan^{-1}(21.85) \approx 87.5° \)

### Final Result
The resultant vector has a magnitude of approximately **13.0 m** and makes an angle of approximately **87.5°** with the X-axis.

---