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

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 (x1, y1, z1) and (x2, y2, z2) is (x2–x1, y2–y1, z2–z1).

Let us assume direction ratios for AB is (r1, r2, r3) and BC is (r4, r5, r6).

The proportional condition can be stated as .

Let us find the direction ratios of AB

(r1, r2, r3) = (1–2, –2–3, 3–(–4))

(r1, r2, r3) = (1–2, –2–3, 3+4)

(r1, r2, r3) = (–1, –5, 7)

Let us find the direction ratios of BC

(r4, r5, r6) = (3–1, 8–(–2), –11–3)

(r4, r5, r6) = (3–1, 8+2, –11–3)

(r4, r5, r6) = (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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