Write a unit vector making equal acute angles with the coordinates axes.

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Solution

Suppose $\vec{r}$ makes an angle $α$ with each of the axis $Ο Χ$ , $Ο Υ$ and $Ο Ζ$ .
Then, its direction cosines are $l = \cos α, m = \cos α, n = \cos α$ .
Now,
$Ι^{2} + m^{2} + n^{2} = 1 \Rightarrow l^{2} + l^{2} + l^{2} = 1 [∵ l = m = n] \Rightarrow 3 l^{2} = 1 \Rightarrow l^{2} = \frac{1}{3} \Rightarrow l = \pm \frac{1}{\sqrt{3}}$
Since, the angle is acute Hence, we take only positive value
Therefore, unit vector is $(\frac{1}{\sqrt{3}} \hat{i} + \frac{1}{\sqrt{3}} \hat{j} + \frac{1}{\sqrt{3}} \hat{k}) .$

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