Question

The vector $\overrightarrow{a}$ which is collinear with the vector $\overrightarrow{b}=\hat{i}-3\hat{j}+\hat{k}$ and satisfies the condition $\overrightarrow{a}\cdot \overrightarrow{b}=22$ is

• A. $2\hat{i}+8\hat{j}-2\hat{k}$
• B. $2\hat{i}-6\hat{j}+2\hat{k}$
• C. $2\hat{i}-8\hat{j}-2\hat{k}$
• D. $2\hat{i}-6\hat{j}-2\hat{k}$

Answer 1

The correct option is B $2\hat{i}-6\hat{j}+2\hat{k}$

If $\overrightarrow{a}$ is collinear with $\overrightarrow{b},$ then$\overrightarrow{a}=x\overrightarrow{b}$
here, $x$ is some scalar $\Rightarrow \overrightarrow{a}\cdot \overrightarrow{b}=x\overrightarrow{b}\cdot \overrightarrow{b}=22$
$\Rightarrow x(\hat{i}-3\hat{j}+\hat{k})\cdot (\hat{i}-3\hat{j}+\hat{k})=22$
$\Rightarrow x(1+9+1)=22$
$\Rightarrow 11x=22$
$\Rightarrow x=2$
$\overrightarrow{a}=x\overrightarrow{b}=2(\hat{i}-3\hat{j}+\hat{k})$
$\therefore \overrightarrow{a}=2\hat{i}-6\hat{j}+2\hat{k}$