Consider a line RS with direction cosines l, m, n. Through the origin draw a line parallel to the given line and take a point P(x, y, z) on this line. From P draw a perpendicular PA on the x-axis in following Fig.
Let OP = r. Then cos α = `(OA)/(OP) = x / r . ` This gives x = lr.
Similarly, y = mr and z = nr
Thus `x^2 + y^2 + z^2 = r^2 (l^2 + m^2 + n^2) `
But `x^2 + y^2 + z^2 = r^2`
Hence `l^2 + m^2 + n^2 = 1`
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