If `vec(tau)xxvec(L)=0` for a rigid body, where `vec(tau)=` resultant torque & `vec(L)` =angular momentum about a point and both are non-zero. Then

Options

A. `vec(L)=`constant
B. `|vec(L)|`=constant
C. `|vec(L)|` will increases
D. `|vec(L)|` may increase

Explanation

To solve the problem, we need to analyze the given condition where the cross product of the resultant torque vector (\(\vec{\tau}\)) and the angular momentum vector (\(\vec{L}\)) is equal to zero:

\[ \vec{\tau} \times \vec{L} = 0 \]

### Step-by-Step Solution:

1. **Understanding the Cross Product**:
The cross product \(\vec{\tau} \times \vec{L}\) is defined as:
\[ \vec{\tau} \times \vec{L} = |\vec{\tau}| |\vec{L}| \sin(\theta) \hat{n} \]
where \(\theta\) is the angle between the vectors \(\vec{\tau}\) and \(\vec{L}\), and \(\hat{n}\) is the unit vector perpendicular to the plane formed by \(\vec{\tau}\) and \(\vec{L}\).

2. **Setting the Condition**:
Since we know that \(\vec{\tau} \times \vec{L} = 0\) and both \(\vec{\tau}\) and \(\vec{L}\) are non-zero vectors, the only way for the cross product to be zero is if:
\[ \sin(\theta) = 0 \]

3. **Finding the Angle**:
The equation \(\sin(\theta) = 0\) implies that:
\[ \theta = 0 \text{ or } \theta = \pi \]
This means that the vectors \(\vec{\tau}\) and \(\vec{L}\) are either parallel (same direction) or antiparallel (opposite direction).

4. **Analyzing the Cases**:
- **Case 1: \(\theta = 0\)** (Vectors in the same direction):
- If \(\vec{\tau}\) and \(\vec{L}\) are in the same direction, the angular momentum \(L\) will increase because the torque is acting in the same direction as the angular momentum.

- **Case 2: \(\theta = \pi\)** (Vectors in opposite directions):
- If \(\vec{\tau}\) and \(\vec{L}\) are in opposite directions, the angular momentum \(L\) will decrease because the torque is acting against the angular momentum.

5. **Conclusion**:
Since the problem states that the angular momentum may increase, we conclude that depending on the angle \(\theta\), the angular momentum can either increase or decrease. Thus, the final answer is that the angular momentum may increase or decrease depending on the orientation of the torque relative to the angular momentum.

### Final Answer:
The angular momentum may increase or decrease depending on the angle between the torque and angular momentum vectors.