Q. 115.0( 1 Vote )

Show that the line through points (1,–1,2) and (3,4,–2) is perpendicular to the line through the points (0,3,2) and (3,5,6).

Answer :

Let us denote the points as follows:

A = (1,–1,2)

B = (3,4,–2)

C = (0,3,2)

D = (3,5,6)

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 AB be (r1,r2,r3) and CD 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 AB

(r1,r2,r3) = (3–1, 4–(–1), –2–2)

(r1,r2,r3) = (3–1, 4+1, –2–2)

(r1,r2,r3) = (2,5,–4)

Let’s find the direction ratios for the line CD

(r4,r5,r6) = (3–0, 5–3, 6–2)

(r4,r5,r6) = (3,2,4)

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

r1.r4+r2.r5+r3.r6 = (2×3)+(5×2)+(–4×4)

r1.r4+r2.r5+r3.r6 = 6+10–16

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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