Vector Algebra
Vector triple product; magnitude
nta_pyq_2023_jan
Grade 12
Question:
If $\vec{a}$, $\vec{b}$, $\vec{c}$ are three non-zero vectors and $\hat{n}$ is a unit vector perpendicular to $\vec{c}$ such that $\vec{a} = \alpha\vec{b} - \hat{n}$, $(\alpha \neq 0)$ and $\vec{b}\cdot\vec{c} = 12$, then $|\vec{c}\times(\vec{a}\times\vec{b})|$ is equal to:
Step-by-Step Solution
Key Concept: Expand $\vec{c}\times(\vec{a}\times\vec{b})$ using BAC-CAB, use $\hat{n}\perp\vec{c}$ and $\vec{a}=\alpha\vec{b}-\hat{n}$.
$\vec{c}\times(\vec{a}\times\vec{b}) = (\vec{c}\cdot\vec{b})\vec{a}-(\vec{c}\cdot\vec{a})\vec{b}$. Since $\vec{a}\cdot\vec{c} = \alpha(\vec{b}\cdot\vec{c}) = 12\alpha$: $= 12\vec{a} - 12\alpha\vec{b} = 12(\vec{a}-\alpha\vec{b}) = 12(-\hat{n})$. $|\cdot|=12$. Answer: (3)
Correct Answer: 12