Q. 115.0( 2 Votes )

# Find the direction cosines of the line passing through two points (–2,4,–5) and (1,2,3).

Answer :

Let us assume the given two points of line be X(–2,4,–5) and Y(1,2,3).

Let us also assume the direction ratios for the given line be (r_{1}, r_{2}, r_{3}).

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}).

So, using this property the direction ratios for the given line is, ⇒ (r_{1}, r_{2}, r_{3}) = (1–(–2), 2–4, 3–(–5))

⇒ (r_{1}, r_{2}, r_{3}) = (1+2, 2–4, 3+5)

⇒ (r_{1}, r_{2}, r_{3}) = (3, –2, 8)

Let us assume be the direction cosines of the given line.

We know that for a line of direction ratios r_{1}, r_{2}, r_{3} and having direction cosines has the following property.

⇒

⇒

⇒

Let us substitute the values of r_{1}, r_{2}, r_{3} to find the values of l, m, n.

⇒

⇒

⇒

⇒

⇒

⇒

⇒

⇒

⇒

∴ The Direction Cosines for the given line is .

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