Q. 45.0( 2 Votes )

# Using direction ratios show that the points A(2,3,–4), B(1,–2,3), C(3,8,–11) are collinear.

Answer :

Given points are:

⇒ A = (2,3,–4)

⇒ B = (1,–2,3)

⇒ C = (3,8,–11)

We know that for points D, E, F to be collinear the direction ratios of any two lines from DE, DF, EF are to be proportional;

We know that direction ratios for a line passing through points (x_{1}, y_{1}, z_{1}) and (x_{2}, y_{2}, z_{2}) is (x_{2}–x_{1}, y_{2}–y_{1}, z_{2}–z_{1}).

Let us assume direction ratios for AB is (r_{1}, r_{2}, r_{3}) and BC is (r_{4}, r_{5}, r_{6}).

The proportional condition can be stated as .

Let us find the direction ratios of AB

⇒ (r_{1}, r_{2}, r_{3}) = (1–2, –2–3, 3–(–4))

⇒ (r_{1}, r_{2}, r_{3}) = (1–2, –2–3, 3+4)

⇒ (r_{1}, r_{2}, r_{3}) = (–1, –5, 7)

Let us find the direction ratios of BC

⇒ (r_{4}, r_{5}, r_{6}) = (3–1, 8–(–2), –11–3)

⇒ (r_{4}, r_{5}, r_{6}) = (3–1, 8+2, –11–3)

⇒ (r_{4}, r_{5}, r_{6}) = (2, 10, –14)

Now

⇒ ……(1)

⇒

⇒ ……(2)

⇒

⇒ ……(3)

From (1),(2),(3) we get,

⇒

So, from the above relational we can say that points A, B , C are collinear.

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