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Q. The unit vector which is orthogonal to the vector $3 i +2 j +6 k$ and is coplanar with the vectors $2 i + j + k$ and $i - j + k$ is

BITSATBITSAT 2007

Solution:

As we know, a vector caplanar to $a, b$ and orthogonalto $c$ is $\lambda\{(a \times b) \times c\}$.
$\therefore $ A vector coplanar to $(2 i + j + k ),( i - j + k )$ and orthogonal to $(3 i +2 j +6 k )$.
$=\lambda[\{2 i + j + k ) \times( i - j + k )\} \times(3 i +2 j +6 k )]$
$=\lambda-21 j +7 k$
$\therefore $ A unit vector is $\pm \frac{ a \times b \times c }{ a \times b \times c }$
$=\pm \frac{-21 j +7 k }{-21^{2}+7^{2}}=\frac{\pm 3 j - k }{\overline{10}}$