If `vec(a)` is a non zero vector of magnitude 'a' and `lambda` a non zero scalar, then `lambda vec(a)` is unit vector if

Options

A. `lambda=1`
B. `lambda=-1`
C. `a=abs(lambda)`
D. `a=1/abs(lambda)`

Answer

To solve the problem, we need to determine under what conditions the vector \( \lambda \vec{a} \) becomes a unit vector. A unit vector has a magnitude of 1.

### Step-by-Step Solution:

1. **Understand the Definitions**:
- A vector \( \vec{a} \) has a magnitude denoted as \( |\vec{a}| = a \).
- A unit vector has a magnitude of 1.

2. **Set Up the Equation**:
- We want to find when \( \lambda \vec{a} \) is a unit vector. This means we need to set the magnitude of \( \lambda \vec{a} \) equal to 1:
\[ |\lambda \vec{a}| = 1 \]

3. **Apply the Properties of Magnitude**:
- The magnitude of a scalar multiplied by a vector is the absolute value of the scalar multiplied by the magnitude of the vector:
\[ |\lambda \vec{a}| = |\lambda| \cdot |\vec{a}| \]

4. **Substitute the Known Magnitude**:
- Since we know \( |\vec{a}| = a \), we can substitute this into our equation:
\[ |\lambda| \cdot a = 1 \]

5. **Solve for \( |\lambda| \)**:
- Rearranging the equation gives us:
\[ |\lambda| = \frac{1}{a} \]

6. **Conclusion**:
- Therefore, \( \lambda \vec{a} \) is a unit vector if:
\[ a = \frac{1}{|\lambda|} \]

### Final Answer:
The condition for \( \lambda \vec{a} \) to be a unit vector is:
\[ a = \frac{1}{|\lambda|} \]