To solve the equation \( \alpha (\vec{a} \times \vec{b}) + \beta (\vec{b} \times \vec{c}) + \lambda (\vec{c} \times \vec{a}) = 0 \), we will follow these steps:
### Step 1: Take the Dot Product with \(\vec{c}\)
We start by taking the dot product of the entire equation with the vector \(\vec{c}\):
\[
\alpha (\vec{a} \times \vec{b}) \cdot \vec{c} + \beta (\vec{b} \times \vec{c}) \cdot \vec{c} + \lambda (\vec{c} \times \vec{a}) \cdot \vec{c} = 0
\]
### Step 2: Simplify the Dot Products
Next, we simplify each term:
- The term \((\vec{b} \times \vec{c}) \cdot \vec{c} = 0\) because the cross product is perpendicular to both \(\vec{b}\) and \(\vec{c}\).
- Similarly, \((\vec{c} \times \vec{a}) \cdot \vec{c} = 0\) for the same reason.
Thus, we have:
\[
\alpha (\vec{a} \times \vec{b}) \cdot \vec{c} = 0
\]
### Step 3: Analyze the Result
The equation \(\alpha (\vec{a} \times \vec{b}) \cdot \vec{c} = 0\) implies that either:
1. \(\alpha = 0\)
2. \((\vec{a} \times \vec{b}) \cdot \vec{c} = 0\)
### Step 4: Take the Dot Product with \(\vec{b}\)
Now, we take the dot product of the original equation with \(\vec{b}\):
\[
\alpha (\vec{a} \times \vec{b}) \cdot \vec{b} + \beta (\vec{b} \times \vec{c}) \cdot \vec{b} + \lambda (\vec{c} \times \vec{a}) \cdot \vec{b} = 0
\]
Again, \((\vec{a} \times \vec{b}) \cdot \vec{b} = 0\) and \((\vec{b} \times \vec{c}) \cdot \vec{b} = 0\). Thus, we have:
\[
\lambda (\vec{c} \times \vec{a}) \cdot \vec{b} = 0
\]
### Step 5: Analyze the Result Again
This implies either:
1. \(\lambda = 0\)
2. \((\vec{c} \times \vec{a}) \cdot \vec{b} = 0\)
### Step 6: Take the Dot Product with \(\vec{a}\)
Finally, we take the dot product of the original equation with \(\vec{a}\):
\[
\alpha (\vec{a} \times \vec{b}) \cdot \vec{a} + \beta (\vec{b} \times \vec{c}) \cdot \vec{a} + \lambda (\vec{c} \times \vec{a}) \cdot \vec{a} = 0
\]
Again, \((\vec{a} \times \vec{b}) \cdot \vec{a} = 0\) and \((\vec{c} \times \vec{a}) \cdot \vec{a} = 0\). Thus, we have:
\[
\beta (\vec{b} \times \vec{c}) \cdot \vec{a} = 0
\]
### Step 7: Final Analysis
This implies either:
1. \(\beta = 0\)
2. \((\vec{b} \times \vec{c}) \cdot \vec{a} = 0\)
### Conclusion
From the three conditions derived, we can conclude that either \(\alpha\), \(\beta\), and \(\lambda\) are all zero, or the scalar triple product \( \vec{a} \cdot (\vec{b} \times \vec{c}) = 0\), which indicates that the vectors \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\) are coplanar.
Since the problem states that \(\alpha\), \(\beta\), and \(\lambda\) are not all zero, we conclude that:
\[
\vec{a} \cdot (\vec{b} \times \vec{c}) = 0
\]
This means that the volume of the parallelepiped formed by the vectors is zero, indicating that the vectors are coplanar.
