Question

If the scalar triple product of three vectors \( \vec{a}, \vec{b}, \vec{c} \) is given as \([\vec{a}\ \vec{b}\ \vec{c}] = 3\), then find the value of the scalar triple product of their cross products, denoted as \([ \vec{a} \times \vec{b} \quad \vec{b} \times \vec{c} \quad \vec{c} \times \vec{a} ]\).

• A. 3
• B. 6
• C. 9
• D. 27

Hint:

Memorize the identity: \[ [\vec{a} \times \vec{b}\ \vec{b} \times \vec{c}\ \vec{c} \times \vec{a}] = [\vec{a}\ \vec{b}\ \vec{c}]^2 \] It saves a lot of time in competitive exams.

Answer 1

The Correct Option is C

Concept: The scalar triple product (STP) of three vectors is defined as: \[ [\vec{u}\ \vec{v}\ \vec{w}] = \vec{u} \cdot (\vec{v} \times \vec{w}) \] Key properties:
• Cyclic property: \[ [\vec{a}\ \vec{b}\ \vec{c}] = [\vec{b}\ \vec{c}\ \vec{a}] = [\vec{c}\ \vec{a}\ \vec{b}] \]
• If two vectors are same/parallel, STP = 0.
• Important identity: \[ [\vec{a} \times \vec{b}\ \ \vec{b} \times \vec{c}\ \ \vec{c} \times \vec{a}] = [\vec{a}\ \vec{b}\ \vec{c}]^2 \]

Step 1: Recognizing required identity.
We need to evaluate: \[ [\vec{a} \times \vec{b}\ \ \vec{b} \times \vec{c}\ \ \vec{c} \times \vec{a}] \] Using the standard vector identity: \[ [\vec{a} \times \vec{b}\ \ \vec{b} \times \vec{c}\ \ \vec{c} \times \vec{a}] = [\vec{a}\ \vec{b}\ \vec{c}]^2 \]

Step 2: Substituting given value.
Given: \[ [\vec{a}\ \vec{b}\ \vec{c}] = 3 \] So, \[ [\vec{a} \times \vec{b}\ \ \vec{b} \times \vec{c}\ \ \vec{c} \times \vec{a}] = 3^2 \]

Step 3: Final calculation.
\[ = 9 \]