If a, b, c, d lie in the same plane then `(a times b) times (c times d)` is equal to

Options

A. c + d
B. 0
C. [a, b, c] a + 2b
D. [b, c, d] c + d

Answer

To solve the problem, we need to analyze the expression \((\mathbf{a} \times \mathbf{b}) \times (\mathbf{c} \times \mathbf{d})\) given that vectors \(\mathbf{a}, \mathbf{b}, \mathbf{c}, \mathbf{d}\) lie in the same plane.

### Step-by-Step Solution:

1. **Understanding the Cross Product**:
The cross product of two vectors results in a vector that is perpendicular to the plane formed by the two vectors. Therefore, \(\mathbf{a} \times \mathbf{b}\) will be a vector perpendicular to the plane containing \(\mathbf{a}\) and \(\mathbf{b}\), and \(\mathbf{c} \times \mathbf{d}\) will be a vector perpendicular to the plane containing \(\mathbf{c}\) and \(\mathbf{d}\).

2. **Vectors in the Same Plane**:
Since all four vectors \(\mathbf{a}, \mathbf{b}, \mathbf{c}, \mathbf{d}\) lie in the same plane, the vectors \(\mathbf{a} \times \mathbf{b}\) and \(\mathbf{c} \times \mathbf{d}\) will be perpendicular to the same plane. This means that both resulting vectors from the cross products are in the same direction or opposite directions.

3. **Calculating the Cross Product**:
The expression \((\mathbf{a} \times \mathbf{b}) \times (\mathbf{c} \times \mathbf{d})\) can be evaluated using the vector triple product identity:
\[ \mathbf{x} \times (\mathbf{y} \times \mathbf{z}) = (\mathbf{x} \cdot \mathbf{z}) \mathbf{y} - (\mathbf{x} \cdot \mathbf{y}) \mathbf{z} \]
Here, let \(\mathbf{x} = \mathbf{a} \times \mathbf{b}\), \(\mathbf{y} = \mathbf{c}\), and \(\mathbf{z} = \mathbf{d}\).

4. **Applying the Triple Product Identity**:
Since \(\mathbf{a} \times \mathbf{b}\) is perpendicular to the plane of \(\mathbf{a}\) and \(\mathbf{b}\), and \(\mathbf{c} \times \mathbf{d}\) is also perpendicular to the plane of \(\mathbf{c}\) and \(\mathbf{d}\), the angle between \(\mathbf{a} \times \mathbf{b}\) and \(\mathbf{c} \times \mathbf{d}\) is either \(0\) or \(\pi\) (they are either in the same direction or opposite directions).

5. **Conclusion**:
Since the sine of \(0\) or \(\pi\) is \(0\), we conclude that:
\[ (\mathbf{a} \times \mathbf{b}) \times (\mathbf{c} \times \mathbf{d}) = \mathbf{0} \]
Thus, the final answer is:
\[ \boxed{0} \]