If vectors
Question:

If vectors $\overrightarrow{\mathrm{a}}_{1}=\mathrm{x} \hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{a}}_{2}=\hat{\mathrm{i}}+\mathrm{y} \hat{\mathrm{j}}+\mathrm{z} \hat{\mathrm{k}}$ are collinear, then a possible unit vector parallel to the vector $x \hat{i}+y \hat{j}+z \hat{k}$ is

  1. $\frac{1}{\sqrt{2}}(-\hat{\mathrm{j}}+\hat{\mathrm{k}})$

  2. $\frac{1}{\sqrt{2}}(\hat{i}-\hat{j})$

  3. $\frac{1}{\sqrt{3}}(\hat{i}+\hat{j}-\hat{k})$

  4. $\frac{1}{\sqrt{3}}(\hat{i}-\hat{j}+\hat{k})$


Correct Option: , 4

Solution:

$\vec{a}_{1}$ and $\vec{a}_{2}$ are collinear

so $\frac{x}{1}=\frac{-1}{y}=\frac{1}{z}$

unit vector in direction of

$x \hat{i}+y \hat{j}+z \hat{k}=\pm \frac{1}{\sqrt{3}}(\hat{i}-\hat{j}+\hat{k})$

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