Question

If vectors ${\overrightarrow{a}}_1=x\hat{i}-\hat{j}+\hat{k}$ and ${\overrightarrow{a}}_2=\hat{i}+y\hat{j}+2\hat{k}$ are collinear, then a possible unit vector parallel to the vector $x\hat{i}+y\hat{j}+z\hat{k}$ is :

• A. $\frac{1}{\sqrt{2}}(-\hat{j}+\hat{k})$
• B. $\frac{1}{\sqrt{2}}(\hat{i}-\hat{j})$
• C. $\frac{1}{\sqrt{3}}(\hat{i}-\hat{j}+\hat{k})$
• D. $\frac{1}{\sqrt{3}}(\hat{i}+\hat{j}-\hat{k})$

Answer 1

The correct option is C $\frac{1}{\sqrt{3}}(\hat{i}-\hat{j}+\hat{k})$

Collinear condition
$\frac{x}{1}=-\frac{1}{y}=\frac{1}{z}=\lambda$ (let)
Unit vector parallel to $x\hat{i}+y\hat{j}+z\hat{k}=\pm \frac{(\lambda \hat{i}-\frac{1}{\lambda }\hat{j}+\frac{1}{\lambda }\hat{k})}{\sqrt{{\lambda }^2+\frac{2}{{\lambda }^2}}}$
For $\lambda =1,$ it is $\pm \frac{(\hat{i}-\hat{j}+\hat{k})}{\sqrt{3}}$