Question

$a =3 \hat{ i }+\hat{ j }-\hat{ k }, b =\hat{ i }-4 \hat{ j }+5 \hat{ k }, c =4 \hat{ i }+5 \hat{ j }-\hat{ k }$ are three vectors and a vector $r$ is perpendicular to both the vectors $b$ and $c$. If $r \cdot a =9$, then $r =$

• A. $3(\hat{ i }-\hat{ j }-\hat{ k })$
• B. $3(\hat{ i }-\hat{ j }+\hat{ k })$
• C. $9(\hat{ i }-\hat{ j }-\hat{ k })$
• D. $9(\hat{ i }-\hat{ j }+\hat{ k })$

Answer 1

Answer: (A)

Given vectors $a =3 \hat{ i }+\hat{ j }-\hat{ k }, b =\hat{ i }-4 \hat{ j }+5 \hat{ k }$
and $c =4 \hat{ i }+5 \hat{ j }-\hat{ k }$
$\because b \times c =\begin{vmatrix}\hat{ i } & \hat{ j } & \hat{ k } \\ 1 & -4 & 5 \\ 4 & 5 & -1\end{vmatrix}=\hat{ i }(4-25)-\hat{ j }(-1-20)$
$+\hat{ k }(5+16)$
$=-21 \hat{ i }+21 \hat{ j }+21 \hat{ k }=21(-\hat{ i }+\hat{ j }+\hat{ k })$
$\Rightarrow | \,b \times c |=21 \sqrt{3}$
Now, vector $r$, which is perpendicular to $b$ and $c$, so
$r =\pm| r | \frac{ b \times c }{| b \times c |}$
$\because \,d r \cdot a = y $
$\Rightarrow \, \pm \frac{| r |}{| b \times c |}[21(-\hat{ i }+\hat{ j }+\hat{ k }) \cdot(3 \hat{ i }+\hat{ j }-\hat{ k })]= 9$
$\Rightarrow \, \pm \frac{| r |}{21 \sqrt{3}}[21(-3+1-1)]=9$
$\Rightarrow \, | r | =3 \sqrt{3} $
$(\because| r |>\,0) $
$\therefore \, r =\pm 3 \sqrt{3} \frac{21(-\hat{ i }+\hat{ j }+\hat{ k })}{21 \sqrt{3}}=\pm 3(-\hat{ i }+\hat{ j }+\hat{ k }) $
$=-3(\hat{ i }+\hat{ j }+ k ) \text { or } 3(\hat{ i }-\hat{ j }-\hat{ k })$