What can you conclude about four non - zero vectors `vec(a), vec(b), vec(c )` and `vec(d)`, given that : `[(vec(a)xx vec(b)).vec(c )]+[(vec(b)xx vec(c )).vec(d)]=0` ?

Answer

To solve the problem, we need to analyze the given expression involving the vectors \(\vec{a}\), \(\vec{b}\), \(\vec{c}\), and \(\vec{d}\):

\[ [(\vec{a} \times \vec{b}) \cdot \vec{c}] + [(\vec{b} \times \vec{c}) \cdot \vec{d}] = 0 \]

### Step 1: Understand the terms involved
The term \((\vec{a} \times \vec{b}) \cdot \vec{c}\) represents the scalar triple product, which can be interpreted as the volume of the parallelepiped formed by the vectors \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\). Similarly, \((\vec{b} \times \vec{c}) \cdot \vec{d}\) represents the volume of the parallelepiped formed by \(\vec{b}\), \(\vec{c}\), and \(\vec{d}\).

### Step 2: Set up the equation
From the equation, we can rewrite it as:

\[ (\vec{a} \times \vec{b}) \cdot \vec{c} = -(\vec{b} \times \vec{c}) \cdot \vec{d} \]

This indicates that the volume represented by the first term is equal in magnitude but opposite in sign to the volume represented by the second term.

### Step 3: Analyze the implications
For the sum of these two volumes to equal zero, it implies that both volumes must be zero. This can only happen if the vectors involved in each of the scalar triple products are coplanar.

### Step 4: Conclusion
Thus, we conclude that the vectors \(\vec{a}\), \(\vec{b}\), \(\vec{c}\), and \(\vec{d}\) must all lie in the same plane. Therefore, we can say:

**Conclusion**: The vectors \(\vec{a}\), \(\vec{b}\), \(\vec{c}\), and \(\vec{d}\) are coplanar.

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