Vector Algebra
Scalar Triple Product
Grade 12
Question:
<p>The value of \([\vec{a}\times\vec{b},\;\vec{b}\times\vec{c},\;\vec{c}\times\vec{a}]\) equals</p>
<li>\([\vec{a},\vec{b},\vec{c}]^3\)</li>
<li>\([\vec{a},\vec{b},\vec{c}]^2\)</li>
<li>\([\vec{a},\vec{b},\vec{c}]\)</li>
<li>\(0\)</li>
Step-by-Step Solution
Key Concept: Use the identity [a \times b, b \times c, c \times a] = [a,b,c]^2. This is a standard scalar triple product identity.
Standard identity: $[\vec{a}\times\vec{b},\;\vec{b}\times\vec{c},\;\vec{c}\times\vec{a}]=[\vec{a},\vec{b},\vec{c}]^2$.
Proof sketch: Each cross product can be expanded using the identity $(\vec{u}\times\vec{v})\cdot(\vec{w}\times\vec{x})=(\vec{u}\cdot\vec{w})(\vec{v}\cdot\vec{x})-(\vec{u}\cdot\vec{x})(\vec{v}\cdot\vec{w})$. After expansion the scalar triple product reduces to $[\vec{a},\vec{b},\vec{c}]^2$.
Answer: (B)
Correct Answer: B