If l, m, n are direction cosines of a line then `l hat(i) + m hat(j) + n hat(k)` is________

Options

A. Null vector
B. the unit vector along the line.
C. Any vector along the line
D. a vector perpendicular to the line

Answer

To solve the problem, we need to analyze the expression given in the question, which is \( l \hat{i} + m \hat{j} + n \hat{k} \), where \( l, m, n \) are direction cosines of a line.

### Step-by-step Solution:

1. **Understanding Direction Cosines**:
- Direction cosines \( l, m, n \) are the cosines of the angles that a line makes with the coordinate axes. They satisfy the equation:
\[ l^2 + m^2 + n^2 = 1 \]

2. **Defining the Vector**:
- We can define a vector \( \vec{a} \) as:
\[ \vec{a} = l \hat{i} + m \hat{j} + n \hat{k} \]

3. **Calculating the Magnitude of the Vector**:
- The magnitude of vector \( \vec{a} \) is given by:
\[ |\vec{a}| = \sqrt{l^2 + m^2 + n^2} \]

4. **Using the Property of Direction Cosines**:
- Since \( l, m, n \) are direction cosines, we can substitute the property:
\[ |\vec{a}| = \sqrt{1} = 1 \]

5. **Conclusion**:
- Since the magnitude of the vector \( \vec{a} \) is 1, this means that \( \vec{a} \) is a unit vector. Therefore, we can conclude that:
\[ l \hat{i} + m \hat{j} + n \hat{k} \text{ is a unit vector along the line.} \]

### Final Answer:
Thus, \( l \hat{i} + m \hat{j} + n \hat{k} \) is a **unit vector** along the line.

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