The torque `tau` on a body about a given point is found to be equal to AxxL where A is a constant vector, and L is the angular momentum of the body about that point. From this it follows that

Options

A. `dL//dt` is perpendiclar to `L` at all instants of time
B. the component of `L` in the direction of `A` does not change with time
C. magnitude of `L` does not change with time
D. `L` does not change with time

Explanation

To solve the problem, we need to analyze the relationship between torque, angular momentum, and the given constant vector \( A \).

### Step-by-Step Solution:

1. **Understanding Torque and Angular Momentum**:
The torque \( \tau \) on a body about a given point is defined as the rate of change of angular momentum \( L \) with respect to time. Mathematically, this is expressed as:
\[ \tau = \frac{dL}{dt} \]

2. **Given Relationship**:
The problem states that the torque is given by:
\[ \tau = A \times L \]
where \( A \) is a constant vector and \( L \) is the angular momentum vector.

3. **Perpendicularity of Torque and Angular Momentum**:
Since torque \( \tau \) is given by the cross product \( A \times L \), it follows that \( \tau \) is perpendicular to both \( A \) and \( L \). This implies:
\[ \tau \perp L \]
Therefore, the rate of change of angular momentum \( \frac{dL}{dt} \) is also perpendicular to \( L \) at all times.

4. **Component of Angular Momentum in Direction of \( A \)**:
To analyze the component of \( L \) in the direction of \( A \), we can express this component as:
\[ L_{\parallel A} = L \cdot \hat{A} \]
where \( \hat{A} \) is the unit vector in the direction of \( A \).

5. **Differentiating the Component**:
We differentiate this component with respect to time:
\[ \frac{d}{dt}(L \cdot \hat{A}) = \frac{dL}{dt} \cdot \hat{A} + L \cdot \frac{d\hat{A}}{dt} \]
Since \( A \) is constant, \( \frac{d\hat{A}}{dt} = 0 \). Thus, we have:
\[ \frac{d}{dt}(L \cdot \hat{A}) = \frac{dL}{dt} \cdot \hat{A} = 0 \]
This indicates that the component of \( L \) in the direction of \( A \) does not change with time.

6. **Conclusion**:
From the above analysis, we can conclude that:
- The torque \( \tau \) is perpendicular to the angular momentum \( L \) at all times.
- The component of angular momentum \( L \) in the direction of the constant vector \( A \) does not change with time.

### Final Answer:
From the given conditions, we conclude that:
1. \( \tau \perp L \) at all times.
2. The component of \( L \) in the direction of \( A \) remains constant over time.