Add vectors `vecA,vecB and vecC` each having magnitude of 100 unit and inclined to the X-axis at angles `45^@, 135^@ and 315^@` respectively.

Explanation

To solve the problem of adding vectors \(\vec{A}\), \(\vec{B}\), and \(\vec{C}\) with given magnitudes and angles, we will follow these steps:

### Step 1: Determine the components of each vector

1. **Vector \(\vec{A}\)**:
- Magnitude: 100 units
- Angle: \(45^\circ\)
- Components:
\[ \vec{A}_x = 100 \cos(45^\circ) = 100 \cdot \frac{1}{\sqrt{2}} = 50\sqrt{2} \]
\[ \vec{A}_y = 100 \sin(45^\circ) = 100 \cdot \frac{1}{\sqrt{2}} = 50\sqrt{2} \]

2. **Vector \(\vec{B}\)**:
- Magnitude: 100 units
- Angle: \(135^\circ\)
- Components:
\[ \vec{B}_x = 100 \cos(135^\circ) = 100 \cdot \left(-\frac{1}{\sqrt{2}}\right) = -50\sqrt{2} \]
\[ \vec{B}_y = 100 \sin(135^\circ) = 100 \cdot \frac{1}{\sqrt{2}} = 50\sqrt{2} \]

3. **Vector \(\vec{C}\)**:
- Magnitude: 100 units
- Angle: \(315^\circ\)
- Components:
\[ \vec{C}_x = 100 \cos(315^\circ) = 100 \cdot \frac{1}{\sqrt{2}} = 50\sqrt{2} \]
\[ \vec{C}_y = 100 \sin(315^\circ) = 100 \cdot \left(-\frac{1}{\sqrt{2}}\right) = -50\sqrt{2} \]

### Step 2: Sum the components of the vectors

Now we will add the components of the three vectors:

- **Total \(x\)-component**:
\[ R_x = \vec{A}_x + \vec{B}_x + \vec{C}_x = 50\sqrt{2} - 50\sqrt{2} + 50\sqrt{2} = 50\sqrt{2} \]

- **Total \(y\)-component**:
\[ R_y = \vec{A}_y + \vec{B}_y + \vec{C}_y = 50\sqrt{2} + 50\sqrt{2} - 50\sqrt{2} = 50\sqrt{2} \]

### Step 3: Calculate the magnitude of the resultant vector

The magnitude of the resultant vector \(\vec{R}\) can be calculated using the Pythagorean theorem:
\[ R = \sqrt{R_x^2 + R_y^2} = \sqrt{(50\sqrt{2})^2 + (50\sqrt{2})^2} = \sqrt{5000 + 5000} = \sqrt{10000} = 100 \]

### Step 4: Determine the direction of the resultant vector

The direction (angle \(\theta\)) can be found using:
\[ \tan(\theta) = \frac{R_y}{R_x} = \frac{50\sqrt{2}}{50\sqrt{2}} = 1 \]
Thus,
\[ \theta = 45^\circ \]

### Final Result

The resultant vector \(\vec{R}\) has a magnitude of 100 units and is inclined at an angle of \(45^\circ\) to the x-axis.

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