Question

If the vectors \( \vec{a}, \vec{b}, \vec{c} \) are non-coplanar, then \( \dfrac{\left| \vec{a} + 2\vec{b} \;\; \vec{b} + 2\vec{c} \;\; \vec{c} + 2\vec{a} \right|} {\left| \vec{a} \;\; \vec{b} \;\; \vec{c} \right|} \) is

• A. \( 8 \)
• B. \( 3 \)
• C. \( 9 \)
• D. \( 6 \)

Hint:

In determinant problems, any determinant with two identical columns (or rows) is zero.

Answer 1

The Correct Option is C

Step 1: Use determinant properties.
Using linearity of determinants, we expand: \[ \left| \vec{a} + 2\vec{b} \;\; \vec{b} + 2\vec{c} \;\; \vec{c} + 2\vec{a} \right| \] Step 2: Expand column-wise.
\[ = \left| \vec{a} \;\; \vec{b} \;\; \vec{c} \right| + 2\left| \vec{b} \;\; \vec{b} \;\; \vec{c} \right| + 2\left| \vec{a} \;\; \vec{c} \;\; \vec{c} \right| + 4\left| \vec{b} \;\; \vec{c} \;\; \vec{a} \right| \] Step 3: Use determinant properties.
Determinants with two identical vectors are zero. Hence, \[ = 9 \left| \vec{a} \;\; \vec{b} \;\; \vec{c} \right| \] Step 4: Form the ratio.
\[ \frac{9 \left| \vec{a} \;\; \vec{b} \;\; \vec{c} \right|} {\left| \vec{a} \;\; \vec{b} \;\; \vec{c} \right|} = 9 \] Step 5: Conclusion.
The required value is \( 9 \).