The condition `(a.b)^(2)=a^(2)b^(2)` is satisfied when

Options

A. a is parallel to b
B. `aneb`
C. `a.b=1`
D. `abotb`

Explanation

To solve the problem, we need to analyze the condition given in the question:

\[ (a \cdot b)^2 = a^2 b^2 \]

### Step-by-step Solution:

1. **Understanding the Dot Product**:
The dot product of two vectors \( a \) and \( b \) can be expressed as:
\[ a \cdot b = |a| |b| \cos \theta \]
where \( \theta \) is the angle between the two vectors.

2. **Substituting the Dot Product**:
Substitute the expression for the dot product into the equation:
\[ (|a| |b| \cos \theta)^2 = |a|^2 |b|^2 \]

3. **Expanding the Left Side**:
Expanding the left side gives:
\[ |a|^2 |b|^2 \cos^2 \theta = |a|^2 |b|^2 \]

4. **Dividing Both Sides**:
Assuming \( |a| \) and \( |b| \) are not zero, we can divide both sides by \( |a|^2 |b|^2 \):
\[ \cos^2 \theta = 1 \]

5. **Finding the Values of \( \theta \)**:
The equation \( \cos^2 \theta = 1 \) implies:
\[ \cos \theta = \pm 1 \]
This means:
- \( \cos \theta = 1 \) when \( \theta = 0^\circ \) (vectors are parallel)
- \( \cos \theta = -1 \) when \( \theta = 180^\circ \) (vectors are anti-parallel)

6. **Conclusion**:
The condition \( (a \cdot b)^2 = a^2 b^2 \) is satisfied when the vectors \( a \) and \( b \) are either parallel or anti-parallel.

### Final Answer:
The condition \((a \cdot b)^2 = a^2 b^2\) is satisfied when the vectors \( a \) and \( b \) are parallel (\( \theta = 0^\circ\)) or anti-parallel (\( \theta = 180^\circ\)).