Question

The volume of the tetrahedron whose co-terminus edges are $\bar{a}, \bar{b}, \bar{c}$ is 12 cubic units. If the scalar projection of $\bar{a}$ on $\bar{b} \times \bar{c}$ is 4 , then $|\bar{b} \times \bar{c}| =$

• A. 18
• B. $\frac{1}{18}$
• C. 16
• D. $\frac{1}{16}$

Hint:

Volume of the tetrahedron is one sixth of the scalar triple product, and the scalar projection divides the triple product by the length of b cross c.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept
Volume of tetrahedron \(=\frac{1}{6}|[\bar a\ \bar b\ \bar c]|=12\), so \(|\bar a\cdot(\bar b\times\bar c)|=72\).

Step 2: Scalar projection
Scalar projection of \(\bar a\) on \(\bar b\times\bar c\) is \(\dfrac{\bar a\cdot(\bar b\times\bar c)}{|\bar b\times\bar c|}=4\).

Step 3: Calculation
\(\dfrac{72}{|\bar b\times\bar c|}=4\), so \(|\bar b\times\bar c|=18\).

Step 4: Compare the options
The value is 18, the first option.

Final Answer:
\(|\bar b\times\bar c|=18\). \[ \boxed{\text{(A) }18} \]