Prove that 1, 1, 1 cannot be direction cosines of a straight line.

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Solution

Let $1, 1, 1$ be the direction cosines of a straight line. Then
$1^{2} + 1^{2} + 1^{2} = 3$ ≠ 1
Since direction cosines of a line which makes equal angle with the axes must satisfy $l^{2} + m^{2} + n^{2} = 1$ .
Hence $1, 1, 1$ cannot be the direction cosines of a straight line.

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