Q.18 A line passes through the points \((6, -7, -1)\) and \((2, -3, 1)\). Find the direction ratios and the direction cosines of the line. Show that the line does not pass through the origin.
Best answer
Answer: d.r.s \(-4, 4, 2\) (i.e. \(-2, 2, 1\)); d.c.s \(-\dfrac23, \dfrac23, \dfrac13\); the origin does not lie on the line.
Direction ratios: \(2 - 6,\ -3 - (-7),\ 1 - (-1) = -4, 4, 2\), i.e. \(-2, 2, 1\).
\(\sqrt{(-2)^2 + 2^2 + 1^2} = 3\), so the direction cosines are \(-\dfrac23, \dfrac23, \dfrac13\) (or \(\dfrac23, -\dfrac23, -\dfrac13\)).
Equation of the line: \(\dfrac{x - 2}{-2} = \dfrac{y + 3}{2} = \dfrac{z - 1}{1}\).
For the origin \((0, 0, 0)\): \(\dfrac{0 - 2}{-2} = 1\), \(\dfrac{0 + 3}{2} = \dfrac32\), \(\dfrac{0 - 1}{1} = -1\). These are not equal, so the origin does not lie on the line.
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