Question

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

• A. $\frac{2i+6j+k}{\overline{41}}$
• B. $\frac{2i+3j}{\overline{13}}$
• C. $\frac{3j-k}{\overline{10}}$
• D. $\frac{4i+3j-3k}{\overline{34}}$

Answer 1

The correct option is C $\frac{3j-k}{\overline{10}}$

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