Q. 125.0( 1 Vote )

Show that the line joining the origin to the point (2,1,1) is perpendicular to the line determined by the points (3,5,–1) and (4,3,–1).

Let us denote the points as follows:

O = (0,0,0)

A = (2,1,1)

B = (3,5,–1)

C = (4,3,–1)

If two lines of direction ratios (a1,b1,c1) and (a2,b2,c2) are said to be perpendicular to each other. Then the following condition is need to be satisfied:

a1.a2+b1.b2+c1.c2=0 ……(1)

Let us assume the direction ratios for line OA be (r1,r2,r3) and BC be (r4,r5,r6)

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’s find the direction ratios for the line OA

(r1,r2,r3) = (2–0, 1–0, 1–0)

(r1,r2,r3) = (2,1,1)

Let’s find the direction ratios for the line BC

(r4,r5,r6) = (4–3, 3–5, –1–(–1))

(r4,r5,r6) = (4–3, 3–5, –1+1)

(r4,r5,r6) = (1,–2,0)

Let us check whether the lines are perpendicular or not using (1)

r1.r4+r2.r5+r3.r6 = (2×1)+(1×–2)+(1×0)

r1.r4+r2.r5+r3.r6 = 2–2+0

r1.r4+r2.r5+r3.r6 = 0

Since the condition is clearly satisfied, we can say that the given lines are perpendicular to each other.

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