Each of two vectors `vecD _(1) and vecD _(2)` lies along a coordinamte axis in the x-y plane. Each vector has its tail at the origin, and the dot product of the two vectors is `vecD_(1), vecD_(2)=- |vecD_(1)|.|vecD_(2)|` Which is possiblility is correct ?

Options

A. `vecD_(1)and vecD_(2)` both lie along the positive x-axis.
B. `vecD_(1)` lies along the positive x-axis `vecD_(2)` lies along the negative x-axis.
C. `vecD_(1) and vecD_(2)` both lie along the positive y-axis.
D. `vecD_(1)` lies along the negative x-axis `vecD_(2)` lies along the negative y-axis.

Answer

To solve the problem, we need to analyze the given information about the two vectors \( \vec{D}_1 \) and \( \vec{D}_2 \).

### Step-by-Step Solution:

1. **Understanding the Dot Product**:
The dot product of two vectors \( \vec{D}_1 \) and \( \vec{D}_2 \) is given by the formula:
\[ \vec{D}_1 \cdot \vec{D}_2 = |\vec{D}_1| |\vec{D}_2| \cos \theta \]
where \( \theta \) is the angle between the two vectors.

**Hint**: Recall that the dot product can be positive, negative, or zero depending on the angle between the vectors.

2. **Given Condition**:
We are given that:
\[ \vec{D}_1 \cdot \vec{D}_2 = - |\vec{D}_1| |\vec{D}_2| \]
This implies that:
\[ |\vec{D}_1| |\vec{D}_2| \cos \theta = - |\vec{D}_1| |\vec{D}_2| \]
From this, we can deduce that:
\[ \cos \theta = -1 \]

**Hint**: When \( \cos \theta = -1 \), it indicates that the angle \( \theta \) is 180 degrees.

3. **Interpreting the Angle**:
An angle of 180 degrees means that the two vectors are pointing in opposite directions. This is characteristic of anti-parallel vectors.

**Hint**: Visualize the vectors on the coordinate axes to see how they can be oriented.

4. **Positioning the Vectors**:
Since both vectors lie along the coordinate axes in the x-y plane and have their tails at the origin, we can conclude:
- If \( \vec{D}_1 \) points in the positive x-direction, then \( \vec{D}_2 \) must point in the negative x-direction.
- If \( \vec{D}_1 \) points in the positive y-direction, then \( \vec{D}_2 \) must point in the negative y-direction.

**Hint**: Draw a diagram to visualize the vectors along the axes.

5. **Possible Configurations**:
Based on the above analysis, the possible configurations of the vectors are:
- \( \vec{D}_1 \) along the positive x-axis and \( \vec{D}_2 \) along the negative x-axis.
- \( \vec{D}_1 \) along the positive y-axis and \( \vec{D}_2 \) along the negative y-axis.

**Hint**: Check the options provided in the question to see which configurations match the analysis.

6. **Conclusion**:
The correct possibility is that \( \vec{D}_1 \) and \( \vec{D}_2 \) are anti-parallel vectors. Thus, they can either be:
- \( \vec{D}_1 \) along the positive x-axis and \( \vec{D}_2 \) along the negative x-axis, or
- \( \vec{D}_1 \) along the positive y-axis and \( \vec{D}_2 \) along the negative y-axis.

**Hint**: Look for the configurations that reflect this anti-parallel nature.