Question:
Find the value of $x$ for which $x(\hat{i}+\hat{j}+\hat{k})$ is a unit vector.
Solution:
$x(\hat{i}+\hat{j}+\hat{k})$ is a unit vector if $|x(\hat{i}+\hat{j}+\hat{k})|=1$
Now,
$|x(\hat{i}+\hat{j}+\hat{k})|=1$
$\Rightarrow \sqrt{x^{2}+x^{2}+x^{2}}=1$
$\Rightarrow \sqrt{3 x^{2}}=1$
$\Rightarrow \sqrt{3} x=1$
$\Rightarrow x=\pm \frac{1}{\sqrt{3}}$
Hence, the required value of $x$ is $\pm \frac{1}{\sqrt{3}}$.
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