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Question
Show that the direction cosines of a vector equally inclined to the axes OX, OY and OZ are \[\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}} .\]
Solution
Suppose the vector makes equal angle \[\alpha\] with the coordinate axis.
Then, its direction cosines are \[l = \cos \alpha , m = \cos \alpha , n = \cos \alpha\].
Therefore,
\[l = m = n\]
\[l^2 + m^2 + n^2 = 1\]
\[ \Rightarrow l^2 + l^2 + l^2 = 1\]
\[ \Rightarrow 3 l^2 = 1\]
\[ \Rightarrow l^2 = \frac{1}{3}\]
\[ \Rightarrow l = \frac{1}{\sqrt{3}}\]
Hence, direction cosines are \[\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}} , \frac{1}{\sqrt{3}}\]
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