Question

\(\vec{a},\vec{b},\vec{c}\) represent three concurrent edges of a rectangular parallelopiped whose lengths are \(5,2\) and \(3\) units respectively, then \((\vec{a}+\vec{b}+\vec{c})\cdot (\vec{a}\times \vec{b}+\vec{b}\times \vec{c}+\vec{c}\times \vec{a})=\)

• A. \(10\)
• B. \(90\)
• C. \(60\)
• D. \(30\)

Hint:

The edge vectors are mutually perpendicular, so only the scalar triple product terms survive.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept
Concurrent edges of a rectangular parallelepiped are mutually perpendicular, so \(\vec{a}\cdot\vec{b}=0\) and so on. Also \(\vec{a}\times\vec{b}\) is parallel to \(\vec{c}\).

Step 2: Expand
\((\vec{a}+\vec{b}+\vec{c})\cdot(\vec{a}\times\vec{b}+\vec{b}\times\vec{c}+\vec{c}\times\vec{a})\).
Terms such as \(\vec{a}\cdot(\vec{a}\times\vec{b})\) vanish.

Step 3: Surviving terms
Only \(\vec{c}\cdot(\vec{a}\times\vec{b})\), \(\vec{a}\cdot(\vec{b}\times\vec{c})\) and \(\vec{b}\cdot(\vec{c}\times\vec{a})\) remain. Each equals \([\vec{a}\,\vec{b}\,\vec{c}]\).

Step 4: Calculation
The sum is \(3[\vec{a}\,\vec{b}\,\vec{c}]=3\times(5\times2\times3)=3\times30=90\).

Final Answer:
The value of the expression is 90. \[ \boxed{\text{(B) }90} \]