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 the vectors \(\vec{A}\), \(\vec{B}\), and \(\vec{C}\) with magnitudes of 100 units and angles of \(45^\circ\), \(135^\circ\), and \(315^\circ\) respectively, we can follow these steps:

### Step 1: Break down each vector into its components
We will use the formula for the components of a vector:
\[ \vec{V} = V_x \hat{i} + V_y \hat{j} \]
where \(V_x = V \cos(\theta)\) and \(V_y = V \sin(\theta)\).

1. **For \(\vec{A}\)**:
- Magnitude = 100 units, Angle = \(45^\circ\)
\[ A_x = 100 \cos(45^\circ) = 100 \times \frac{\sqrt{2}}{2} = 50\sqrt{2} \]
\[ A_y = 100 \sin(45^\circ) = 100 \times \frac{\sqrt{2}}{2} = 50\sqrt{2} \]
Thus, \(\vec{A} = 50\sqrt{2} \hat{i} + 50\sqrt{2} \hat{j}\).

2. **For \(\vec{B}\)**:
- Magnitude = 100 units, Angle = \(135^\circ\)
\[ B_x = 100 \cos(135^\circ) = 100 \times -\frac{\sqrt{2}}{2} = -50\sqrt{2} \]
\[ B_y = 100 \sin(135^\circ) = 100 \times \frac{\sqrt{2}}{2} = 50\sqrt{2} \]
Thus, \(\vec{B} = -50\sqrt{2} \hat{i} + 50\sqrt{2} \hat{j}\).

3. **For \(\vec{C}\)**:
- Magnitude = 100 units, Angle = \(315^\circ\)
\[ C_x = 100 \cos(315^\circ) = 100 \times \frac{\sqrt{2}}{2} = 50\sqrt{2} \]
\[ C_y = 100 \sin(315^\circ) = 100 \times -\frac{\sqrt{2}}{2} = -50\sqrt{2} \]
Thus, \(\vec{C} = 50\sqrt{2} \hat{i} - 50\sqrt{2} \hat{j}\).

### Step 2: Sum the components of the vectors
Now, we will add the components of the vectors:

- **Sum of x-components**:
\[ R_x = A_x + B_x + C_x = (50\sqrt{2}) + (-50\sqrt{2}) + (50\sqrt{2}) = 50\sqrt{2} \]

- **Sum of y-components**:
\[ R_y = A_y + B_y + C_y = (50\sqrt{2}) + (50\sqrt{2}) + (-50\sqrt{2}) = 50\sqrt{2} \]

### Step 3: Resultant vector
The resultant vector \(\vec{R}\) can be expressed as:
\[ \vec{R} = R_x \hat{i} + R_y \hat{j} = 50\sqrt{2} \hat{i} + 50\sqrt{2} \hat{j} \]

### Step 4: Magnitude of the resultant vector
To find the magnitude of the resultant vector:
\[ |\vec{R}| = \sqrt{R_x^2 + R_y^2} = \sqrt{(50\sqrt{2})^2 + (50\sqrt{2})^2} = \sqrt{2500 \times 2 + 2500 \times 2} = \sqrt{5000} = 100\sqrt{5} \]

### Conclusion
The resultant vector \(\vec{R}\) has a magnitude of \(100\sqrt{5}\) units, and it is directed at an angle \(\theta\) with respect to the x-axis given by:
\[ \tan(\theta) = \frac{R_y}{R_x} = \frac{50\sqrt{2}}{50\sqrt{2}} = 1 \implies \theta = 45^\circ \]

### Final Result
Thus, the resultant vector has a magnitude of \(100\sqrt{5}\) units and is directed at an angle of \(45^\circ\) with respect to the x-axis.

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