When two vectors a and b of magnitudes 'a' and 'b' respectively are added, the magnitude of resultant vector is always

Options

A. Equal to (a + b)
B. Less than (a + b)
C. Greater than (a + b)
D. Not greater than (a + b)

Answer

To solve the question regarding the magnitude of the resultant vector when two vectors \( \mathbf{a} \) and \( \mathbf{b} \) are added, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Vector Addition**:
When two vectors \( \mathbf{a} \) and \( \mathbf{b} \) are added, the resultant vector \( \mathbf{R} \) can be expressed as:
\[ \mathbf{R} = \mathbf{a} + \mathbf{b} \]

2. **Using the Triangle Inequality**:
According to the triangle inequality in vector addition, the magnitude of the resultant vector \( R \) is given by:
\[ |\mathbf{R}| \leq |\mathbf{a}| + |\mathbf{b}| \]
This means that the magnitude of the resultant vector is always less than or equal to the sum of the magnitudes of the individual vectors.

3. **Considering the Magnitudes**:
Let \( |\mathbf{a}| = a \) and \( |\mathbf{b}| = b \). Therefore, we can rewrite the inequality as:
\[ |\mathbf{R}| \leq a + b \]

4. **Finding the Minimum Magnitude**:
The magnitude of the resultant vector can also be at its minimum when the vectors are in opposite directions. In this case, the magnitude of the resultant vector can be expressed as:
\[ |\mathbf{R}| \geq |a - b| \]

5. **Conclusion**:
Therefore, the magnitude of the resultant vector \( |\mathbf{R}| \) will always satisfy the following condition:
\[ |a - b| \leq |\mathbf{R}| \leq a + b \]
This indicates that the magnitude of the resultant vector is always less than or equal to the sum of the magnitudes of the individual vectors.

### Final Answer:
The magnitude of the resultant vector when two vectors \( \mathbf{a} \) and \( \mathbf{b} \) are added is always less than or equal to \( a + b \).