Statement-1: Unit vectors orthogonal to the vector $3\hat i+2\hat j+6\hat k$ and coplanar with the vectors $2\hat i+\hat j+\hat k$ and $\hat i-\hat j+\hat k$ are $±\frac{1}{\sqrt{10}}(3\hat i-\hat k)$.
Statement-2: For any three vectors $\vec a, \vec b$ and $\vec c$, vector $\vec a×(\vec b×\vec c)$ is orthogonal to a and lies in the plane of $\vec b$ and $\vec c$.
Answer & explanation
Correct answer: option 1
Statement-2 is true.
Let $\vec a = 3\hat i+2\hat j+6\hat k, \vec b=2\hat i+\hat j+\hat k$ and $\vec c=\hat i-\hat j$
Using statement-2, required unit vectors are given by
$α = ±\frac{\vec a×(\vec b×\vec c)}{|\vec a×(\vec b×\vec c)|}$
Now, $\vec a×(\vec b×\vec c)=(\vec a.\vec c)\vec b-(\vec a.\vec b)\vec c$
$⇒\vec a×(\vec b×\vec c)=7(2\hat i+\hat j+\hat k)-14(\hat i-\hat j+\hat k)=21\hat j-7\hat k$
$∴α = ±\frac{21\hat j-7\hat k}{7\sqrt{10}}=±\frac{1}{\sqrt{10}}(3\hat i-\hat k)$
So, statement-1 is true and statement-2 is a correct explanation of statement-1.