Question

Find the angle between the pairs of lines with direction ratios proportional to a,b,c and (b-c) , (c-a) , (a-b)


Ans:- π/2

Answer 1

$\boxed{\underline{\textsf{Formula :}}}$

If l, m, n and l', m', n' be the direction ratios of two lines, and θ be the angle between those two lines, we can show that

cosθ = $\frac{ll' + mm' + nn'}{\sqrt{l^{2}+m^{2}+n^{2}}\sqrt{l'^{2}+m'^{2}+n'^{2}}}$

$\boxed{\underline{\textsf{Solution :}}}$

Direction ratios of the two lines are proportional to a, b, c and (b - c), (c - a), (a - b)

If the required angle between those two lines be θ

cosθ = $\frac{a (b-c)+b (c-a)+c (a-b)}{\sqrt{a^{2}+b^{2}+c^{2}}\sqrt{(a-b)^{2}+(b-c)^{2}+(c-a)^{2}}}$

= $\frac{ab-ca+bc-ab+ca-bc}{\sqrt{a^{2}+b^{2}+c^{2}}\sqrt{(a-b)^{2}+(b-c)^{2}+(c-a)^{2}}}$

= $\frac{0}{\sqrt{a^{2}+b^{2}+c^{2}}\sqrt{(a-b)^{2}+(b-c)^{2}+(c-a)^{2}}}$

= 0

⇒ cosθ = 0

⇒ θ = $\frac{\pi}{2}$

Hence, the given lines make right angle between them and thus we can conclude that the two lines are perpendicular to each other.

Answer 2

∅=π/2

is the correct answer