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 (a_{1},b_{1},c_{1}) and (a_{2},b_{2},c_{2}) are said to be perpendicular to each other. Then the following condition is need to be satisfied:

⇒ a_{1}.a_{2}+b_{1}.b_{2}+c_{1}.c_{2}=0 ……(1)

Let us assume the direction ratios for line AB be (r_{1},r_{2},r_{3}) and CD be (r_{4},r_{5},r_{6})

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’s find the direction ratios for the line AB

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

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

⇒ (r_{1},r_{2},r_{3}) = (2,5,–4)

Let’s find the direction ratios for the line CD

⇒ (r_{4},r_{5},r_{6}) = (3–0, 5–3, 6–2)

⇒ (r_{4},r_{5},r_{6}) = (3,2,4)

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

⇒ r_{1}.r_{4}+r_{2}.r_{5}+r_{3}.r_{6} = (2×3)+(5×2)+(–4×4)

⇒ r_{1}.r_{4}+r_{2}.r_{5}+r_{3}.r_{6} = 6+10–16

⇒ r_{1}.r_{4}+r_{2}.r_{5}+r_{3.}r_{6} = 0

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

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