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).

Answer :

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 (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 OA be (r_{1},r_{2},r_{3}) and BC 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 OA

⇒ (r_{1},r_{2},r_{3}) = (2–0, 1–0, 1–0)

⇒ (r_{1},r_{2},r_{3}) = (2,1,1)

Let’s find the direction ratios for the line BC

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

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

⇒ (r_{4},r_{5},r_{6}) = (1,–2,0)

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

⇒ r_{1}.r_{4}+r_{2}.r_{5}+r_{3}.r_{6} = (2×1)+(1×–2)+(1×0)

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

⇒ 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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