[a b c] is a scalar triple product of three vectros a⃗, b⃗ and c⃗ then [a b c] is equal to (a) [

step 1 Here we have the scalar triple product of the vectors A, B, and C. So by definition, the scalar

[a b c] is a scalar triple product of three vectros a⃗, b⃗...

$[\overrightarrow{a b c}]$ is a scalar triple product of three vectros $\vec{a}, \vec{b}$ and $\vec{c}$ then $[\overrightarrow{a b c}]$ is equal to
(a) $[\vec{c} \vec{b} a]$
(b) $[\overrightarrow{c a b}]$
(c) $[\overrightarrow{a c b}]$
(d) $[\vec{b} \overrightarrow{a c}]$

Key Concepts

Parity of Permutations

The parity of permutations refers to whether a permutation is even or odd. An even permutation (which can be achieved by any cyclic rearrangement) leaves the scalar triple product unchanged, while an odd permutation (which involves swapping two vectors) changes its sign. Understanding these properties helps in determining the equality or sign difference between different arrangements of vectors in scalar triple products.

Cyclic Permutations

Cyclic permutations of the vectors in a scalar triple product do not change its value. This means that if you have [a, b, c], then [b, c, a] and [c, a, b] will all have the same value. This concept is critical when comparing different arrangements of the same vectors in a scalar triple product.

Scalar Triple Product

The scalar triple product is an operation on three vectors, typically denoted as [a, b, c] or a · (b × c), that results in a scalar. This product geometrically represents the volume of the parallelepiped defined by the three vectors and can be positive, negative, or zero depending on the orientation and coplanarity of the vectors.

Permutation Properties

The scalar triple product has special properties under permutation of its vectors. In particular, cyclic permutations (rotating the order of the vectors) leave the value unchanged, while switching any two vectors (an odd permutation) reverses the sign of the product. These properties are essential in simplifying expressions involving the scalar triple product.

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