Vectors
Vectors
Allen Star Batch
Grade 12
Question:
If $4\vec{a} + 5\vec{b} + 9\vec{c} = \vec{0}$, then $(\vec{a} \times \vec{b}) \times (\vec{b} \times \vec{c}) \times (\vec{c} \times \vec{a})$ is equal to:
A vector perpendicular to plane of $\vec{a}, \vec{b}$ and $\vec{c}$
A scalar quantity
$\vec{0}$
None of these
Step-by-Step Solution
Key Concept: Coplanar vectors have cross products that are parallel (collinear), making their cross product zero.
Given $4\vec{a} + 5\vec{b} + 9\vec{c} = 0$, the vectors $\vec{a}$, $\vec{b}$, $\vec{c}$ are coplanar. This means $\vec{b} \times \vec{c}$ and $\vec{c} \times \vec{a}$ are collinear, so $(\vec{b} \times \vec{c}) \times (\vec{c} \times \vec{a}) = \vec{0}$.
Correct Answer: 3