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Question
Write a unit vector making equal acute angles with the coordinates axes.
Solution
Suppose \[\vec{r}\] makes an angle \[\alpha\] with each of the axis \[OX\], \[O\Upsilon\] and \[OZ\].
Then, its direction cosines are \[l = \cos \alpha, m = \cos\alpha, n = \cos\alpha\]
Now,
\[I^2 + m^2 + n^2 = 1\]
\[ \Rightarrow l^2 + l^2 + l^2 = 1 \left[ \because l = m = n \right]\]
\[ \Rightarrow 3 l^2 = 1\]
\[ \Rightarrow l^2 = \frac{1}{3}\]
\[ \Rightarrow l = \pm \frac{1}{\sqrt{3}}\]
Since, the angle is acute Hence, we take only positive value
Therefore, unit vector is \[\left( \frac{1}{\sqrt{3}} \hat{i} + \frac{1}{\sqrt{3}} \hat{j} + \frac{1}{\sqrt{3}} \hat{k} \right) .\]
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