The direction ratios of the line x − y + z − 5 = 0 = x − 3y − 6 are proportional to

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3, 1, −2 We have , x − y + z − 5 = 0 = x − 3y − 6 \Rightarrow x - 3y - 6 = 0 x - y + z - 5 = 0 \Rightarrow x = 3y + 6 . . . \left( 1 \right) x - y + z - 5 = 0 . . . \left( 2 \right) From (1) and (2), we get , 3y + 6 - y + z - 5 = 0 \Rightarrow 2y + z + 1 = 0 \Rightarrow y = - z - $\frac{1}{2}$ y = x - $\frac{6}{3}$ \left[\text { From } \left( 1 \right) \right] \therefore x - $\frac{6}{3}$ = y = - z - $\frac{1}{2}$ So, the given equation can be re-written as x - $\frac{6}{3}$ = y/1 = z + 1/- 2 Hence, the direction ratios of the given line are proportional to 3, 1, -2 .

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