Question

The acute angle between the two lines whose direction ratios are (1, 1, 2) and (√3 − 1, −√3 − 1, 4) is

(A) \(\frac \pi 3\)

(B) \(\frac \pi 4\)

(C) \(\frac \pi 2\)

(D) \(\frac \pi 6\)

Answer 1

Correct option is (A) \(\frac \pi 3\)

Let, (a₁, b₁, c₁) and (a₂, b₂, c₂) be the direction ratio of two line and \(\theta\) be the angle between them, then we know,

\(\cos \theta = \frac{a_1a_2 + b_1 b_2 + c_1c_2}{\sqrt{{a_1}^2 + {b_1}^2 + {c_1}^2}\sqrt{{a_2}^2 + {b_2}^2 + {c_2}^2}}\)

Given direction ratio are (1, 1, 2) and (√3 − 1, −√3 − 1, 4).

Let angle between them is \(\theta\).

\(\cos \theta = \frac{1.(\sqrt 3 -1)+ 1 .(- \sqrt 3 -1) + 2.4}{\sqrt{{1}^2 + {1}^2 + {2}^2}\sqrt{{(\sqrt 3-1)}^2 + {(-\sqrt 3 - 1)}^2 + {4}^2}}\)

\(= \frac{\sqrt 3 - 1 - \sqrt 3 -1 + 8}{\sqrt 6. \sqrt 24}\)

\(= \frac 6{\sqrt{144}}\)

\(= \frac 6{12}\)

\(= \frac 12\)

\(\cos\theta= \frac 12\)

⇒ \(\theta = \frac \pi3\)