Question

If \(\vec{a}, \vec{b}, \vec{c}\) are three non-coplanar vectors, then \([\vec{a}\times\vec{b},\ \vec{b}\times\vec{c},\ \vec{c}\times\vec{a}]\) is equal to:

• A. \([\vec{a}\ \vec{b}\ \vec{c}]^3\)
• B. \([\vec{a}\ \vec{b}\ \vec{c}]^2\)
• C. 0
• D. None of these

Hint:

For non-coplanar vectors, the scalar triple product of cross products equals the square of the original scalar triple product.

Answer 1

The Correct Option is B

The problem asks to find the value of the expression \([\vec{a}\times\vec{b},\ \vec{b}\times\vec{c},\ \vec{c}\times\vec{a}]\) given that \(\vec{a}, \vec{b}, \vec{c}\) are three non-coplanar vectors.

This expression involves the triple product of the cross products of these vectors. Let's break down the solution:

The expression \([\vec{a}\times\vec{b},\ \vec{b}\times\vec{c},\ \vec{c}\times\vec{a}]\) represents the scalar triple product of the cross products of the vectors.

We use the property of scalar triple product which states that if \(\vec{u} = \vec{v} \times \vec{w}\) and \(\vec{x}\) is any vector, then \([\vec{x}, \vec{v}, \vec{w}] = (\vec{x}\cdot(\vec{v} \times \vec{w}))\).

By the cyclic nature of the determinant and properties of cross product, we have:

\([\vec{a}\times\vec{b},\ \vec{b}\times\vec{c},\ \vec{c}\times\vec{a}] = [\vec{a}, \vec{b}, \vec{c}]^2\)

We conclude that the given expression simplifies to the square of the scalar triple product of the original vectors.

Thus, the correct answer is: \([\vec{a}\ \vec{b}\ \vec{c}]^2\)