Question

Let \(\overline{\mathrm{a}}\) and \(\overline{\mathrm{c}}\) be unit vectors at an angle \(\frac{\pi}{3}\) with each other. If \((\overline{\mathrm{a}} \times(\overline{\mathrm{b}} \times \overline{\mathrm{c}})) \cdot(\overline{\mathrm{a}} \times \overline{\mathrm{c}})=5\), then \([\overline{\mathrm{a}} \overline{\mathrm{b}} \overline{\mathrm{c}}]=\)

• A. 10
• B. -10
• C. 9
• D. \(-9\)

Hint:

Expand \(\bar a\times(\bar b\times\bar c)\) with the BAC-CAB rule, then relate the result to \([\bar a\ \bar b\ \bar c]\).

Answer 1

The Correct Option is B

Step 1: Understanding the Concept
\(\bar a\times(\bar b\times\bar c)=\bar b(\bar a\cdot\bar c)-\bar c(\bar a\cdot\bar b)\).

Step 2: Dot with the Second Factor
The \(\bar c\) term gives \(\bar c\cdot(\bar a\times\bar c)=0\). So the product is \((\bar a\cdot\bar c)\,\bar b\cdot(\bar a\times\bar c)\).

Step 3: Use the Angle
\(\bar a\cdot\bar c=\cos\dfrac{\pi}{3}=\dfrac12\). Also \(\bar b\cdot(\bar a\times\bar c)=-[\bar a\ \bar b\ \bar c]\).

Step 4: Calculation
\(\dfrac12\left(-[\bar a\ \bar b\ \bar c]\right)=5\Rightarrow[\bar a\ \bar b\ \bar c]=-10\).

Final Answer:
The scalar triple product equals -10. \[ \boxed{\text{(B) }-10} \]