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