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 find the resultant of the two vectors \(\vec{A}\) and \(\vec{B}\), we will follow these steps:

### Step 1: Identify the Magnitudes and Angles
- The magnitude of vector \(\vec{A}\) is \(3 \, \text{m}\) and it makes an angle of \(20^\circ\) with the X-axis.
- The magnitude of vector \(\vec{B}\) is \(4 \, \text{m}\) and it makes an angle of \(110^\circ\) with the X-axis.

### Step 2: Calculate the Components of Each Vector
We can express each vector in terms of its components along the X and Y axes.

For vector \(\vec{A}\):
- \(A_x = A \cos(20^\circ) = 3 \cos(20^\circ)\)
- \(A_y = A \sin(20^\circ) = 3 \sin(20^\circ)\)

For vector \(\vec{B}\):
- \(B_x = B \cos(110^\circ) = 4 \cos(110^\circ)\)
- \(B_y = B \sin(110^\circ) = 4 \sin(110^\circ)\)

### Step 3: Calculate the Components
Using trigonometric values:
- \(\cos(20^\circ) \approx 0.9397\)
- \(\sin(20^\circ) \approx 0.3420\)
- \(\cos(110^\circ) \approx -0.3420\)
- \(\sin(110^\circ) \approx 0.9397\)

Now substituting these values:
- \(A_x = 3 \times 0.9397 \approx 2.8191 \, \text{m}\)
- \(A_y = 3 \times 0.3420 \approx 1.0260 \, \text{m}\)
- \(B_x = 4 \times (-0.3420) \approx -1.3680 \, \text{m}\)
- \(B_y = 4 \times 0.9397 \approx 3.7588 \, \text{m}\)

### Step 4: Find the Resultant Components
Now, we can find the components of the resultant vector \(\vec{R}\):
- \(R_x = A_x + B_x = 2.8191 - 1.3680 \approx 1.4511 \, \text{m}\)
- \(R_y = A_y + B_y = 1.0260 + 3.7588 \approx 4.7848 \, \text{m}\)

### Step 5: Calculate the Magnitude of the Resultant Vector
The magnitude of the resultant vector \(\vec{R}\) can be calculated using the Pythagorean theorem:
\[ R = \sqrt{R_x^2 + R_y^2} \]
Substituting the values:
\[ R = \sqrt{(1.4511)^2 + (4.7848)^2} \approx \sqrt{2.1077 + 22.9119} \approx \sqrt{25.0196} \approx 5.0 \, \text{m} \]

### Step 6: Conclusion
The magnitude of the resultant vector \(\vec{R}\) is approximately \(5.0 \, \text{m}\).

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