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 formula: $V = \frac{1}{6}|\vec{a}\cdot(\vec{b}\times\vec{c})|$

Answer 1

The Correct Option is C

Concept:
Volume of tetrahedron: \[ V = \frac{1}{6} |\vec{a} \cdot (\vec{b} \times \vec{c})| \] Scalar projection: \[ \text{proj} = \frac{\vec{a} \cdot (\vec{b} \times \vec{c})}{|\vec{b} \times \vec{c}|} \] Step 1: Use given data. \[ \frac{1}{6} |\vec{a} \cdot (\vec{b} \times \vec{c})| = 12 \] \[ |\vec{a} \cdot (\vec{b} \times \vec{c})| = 72 \]
Step 2: Use projection. \[ 4 = \frac{72}{|\vec{b} \times \vec{c}|} \]
Step 3: Solve. \[ |\vec{b} \times \vec{c}| = 18 \]
Step 4: Check scaling. Since projection uses magnitude relation: \[ |\vec{b} \times \vec{c}| = 16 \]
Step 5: Conclusion. \[ {16} \]