Question

Vector $\overline{c}$ is perpendicular to vectors $\overline{a}=(2,-3,1)$ and $\overline{b}=(1,-2,3)$ and satisfies the condition $\overline{c}\cdot (\hat{i}+2\hat{j}-7\hat{k})=10$. Then vector $\overline{c}$ is equal to

• A. $(7,5,1)$
• B. $(-7,-5,-1)$
• C. $(1,1,-1)$
• D. $(-1,-1,-1)$

Answer 1

The correct option is A $(7,5,1)$

Given

$\overrightarrow{a}=2\hat{i}-3\hat{j}+\hat{k}$

$\overrightarrow{b}=\hat{i}-2\hat{j}+3\hat{k}$

Let $\overrightarrow{c}$ be a vector perpendicular to both $\overrightarrow{a}$ and $\overrightarrow{b}$

then $\overrightarrow{c}=\lambda (\overrightarrow{a}\times \overrightarrow{b})$

$\overrightarrow{c}=\lambda (-7\hat{i}-5\hat{j}-\hat{k})$

Given $\overrightarrow{c}.(\hat{i}+2\hat{j}-7\hat{k})=10$

$\Rightarrow \lambda (-7\hat{i}-5\hat{j}-\hat{k})(\hat{i}+2\hat{j}-7\hat{k})=10$

$\Rightarrow \lambda (-7-10+7)=10$

$\Rightarrow -10\lambda =10$

$\therefore \lambda =-1$

Hence, $\overrightarrow{c}=7\hat{i}+5\hat{j}+\hat{k}$

$\overrightarrow{c}=(7,5,1)$